CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber...
Transcript of CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber...
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CS3383 Unit 1: Divide and ConquerIntroduction
David Bremner
January 7, 2018
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unit prereqs
▶ mergesort▶ geometric series (CLRS A.5)
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Contents
Divide and ConquerIntroMerge SortRecursion Tree for recurrencesInteger Multiplication
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Structure of divide and conquer
function Solve(P)if |𝑃 | is small then
SolveDirectly(𝑃)else
𝑃1 … 𝑃𝑘 = Partition(𝑃 )for 𝑖 = 1 … 𝑘 do
𝑆𝑖 = Solve(𝑃𝑖)end forCombine(𝑆1 … 𝑆𝑘)
end ifend function
▶ Where is the actualwork?
▶ How manysubproblems?
▶ How big are thesubproblems?
![Page 5: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/5.jpg)
Structure of divide and conquer
function Solve(P)if |𝑃 | is small then
SolveDirectly(𝑃)else
𝑃1 … 𝑃𝑘 = Partition(𝑃 )for 𝑖 = 1 … 𝑘 do
𝑆𝑖 = Solve(𝑃𝑖)end forCombine(𝑆1 … 𝑆𝑘)
end ifend function
▶ Where is the actualwork?
▶ How manysubproblems?
▶ How big are thesubproblems?
![Page 6: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/6.jpg)
Structure of divide and conquer
function Solve(P)if |𝑃 | is small then
SolveDirectly(𝑃)else
𝑃1 … 𝑃𝑘 = Partition(𝑃 )for 𝑖 = 1 … 𝑘 do
𝑆𝑖 = Solve(𝑃𝑖)end forCombine(𝑆1 … 𝑆𝑘)
end ifend function
▶ Where is the actualwork?
▶ How manysubproblems?
▶ How big are thesubproblems?
![Page 7: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/7.jpg)
Contents
Divide and ConquerIntroMerge SortRecursion Tree for recurrencesInteger Multiplication
![Page 9: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/9.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
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September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 11: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/11.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 12: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/12.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
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20
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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12
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 13: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/13.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 14: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/14.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
11
9
1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
11
9
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1
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2
September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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7
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12
11
9
1
1
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
11
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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1
1
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
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1
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13
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
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7
2
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9
1
1
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
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2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
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2
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 15: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/15.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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12
11
9
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1
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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2
12
11
9
1
1
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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12
11
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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1
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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2
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
13
7
2
12
11
9
1
1
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13
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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11
9
2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
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2
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 16: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/16.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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7
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13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 17: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/17.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 18: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/18.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
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2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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13
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
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7
2
12
11
9
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
11
9
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
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20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 19: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/19.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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12
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
11
9
1
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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2
September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
11
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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1220
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2
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1
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
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9
1
1
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13
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2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
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2
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 20: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/20.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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12
11
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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12
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
13
7
2
12
11
9
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1
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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13
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2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 21: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/21.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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13
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2
12
11
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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11
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1220
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 22: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/22.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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7
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1220
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7
2
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
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1
20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 23: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/23.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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2
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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2
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11
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
13
7
2
12
11
9
1
1
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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13
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2
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2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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9
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12
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 24: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/24.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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1220
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1
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
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9
1
1
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13
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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2
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 25: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/25.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
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2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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12
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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13
7
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
13
7
2
12
11
9
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1
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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13
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2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 26: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/26.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 27: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/27.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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7
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 28: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/28.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
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2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
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13
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
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7
2
12
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9
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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7
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12
11
9
1
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13
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 29: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/29.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
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2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1220
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 30: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/30.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
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2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
11
9
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13
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
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7
2
12
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
11
9
1
1
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13
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 31: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/31.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
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11
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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7
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11
9
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
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12
11
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1
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13
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1220
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 32: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/32.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
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7
2
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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12
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9
1
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 33: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/33.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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12
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1
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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2
12
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9
1
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7
September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
1220
13
7
2
12
11
9
1
1
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13
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2
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
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13
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2
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9
2
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
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9
2
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12
12
Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 34: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/34.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 35: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/35.jpg)
September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists.
Key subroutine: MERGE
September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing merge sort
MERGE-SORT A[1 . . n]1. If n = 1, done.2. Recursively sort A[ 1 . . ⎡n/2⎤ ]
and A[ ⎡n/2⎤+1 . . n ] .3. “Merge” the 2 sorted lists
T(n)Θ(1)2T(n/2)
Θ(n)
Sloppiness: Should be T( ⎡n/2⎤) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;
2T(n/2) + Θ(n) if n > 1.
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
• We will see several ways starting with "Rec. Tree"
to find a good upper bound on T(n).
September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
![Page 36: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/36.jpg)
Contents
Divide and ConquerIntroMerge SortRecursion Tree for recurrencesInteger Multiplication
![Page 37: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/37.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
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September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
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September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 40: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/40.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 41: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/41.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 42: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/42.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 43: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/43.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
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September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
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September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric seriesSeptember 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 46: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/46.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric series
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 47: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/47.jpg)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
Recursion-tree method
• A recursion tree models the costs (time) of a recursive execution of an algorithm.
• The recursion-tree method can be unreliable, just like any method that uses ellipses (…).
• The recursion-tree method promotes intuition, however.
• The recursion tree method is good for generating guesses for the substitution method.
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
T(n/4) T(n/2)
n2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/4)2 (n/2)2
T(n/16) T(n/8) T(n/8) T(n/4)
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
n2 2n
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2 (n/2)2
Θ(1)
…
2165 n
2nn2
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
n2
(n/2)2
…
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
Example of recursion treeSolve T(n) = T(n/4) + T(n/2) + n2:
(n/16)2 (n/8)2 (n/8)2 (n/4)2
(n/4)2
Θ(1)
…
2165 n
2n
225625 n
( ) ( )( ) 1 31652
165
1652 L++++n
= Θ(n2)
…
Total =
n2
(n/2)2
geometric series
September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
Appendix: geometric series
1
111
2x
xxxxn
n−
−=+++++
L for x ≠ 1
1
11 2x
xx−
=+++ L for |x| < 1
Return to last slide viewed.
![Page 48: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/48.jpg)
extra example
𝑇 (𝑛) = 2𝑇 (3𝑛/8) + 𝑛2
▶ board
![Page 49: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/49.jpg)
Contents
Divide and ConquerIntroMerge SortRecursion Tree for recurrencesInteger Multiplication
![Page 50: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/50.jpg)
Integer Multiplication
Consider the problem of multiplying together two arbitrarily largenumbers.
Input: positive integers 𝑥 and 𝑦, each 𝑛 bits longOutput: positive integer 𝑧 where 𝑧 = 𝑥 ⋅ 𝑦
A straightforward approach using base-2 arithmetic, akin to how wemultiply by hand, takes Θ(𝑛2) time.
Could we do better if we used results from subinstances?
![Page 51: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/51.jpg)
Integer Multiplication
Consider the problem of multiplying together two arbitrarily largenumbers.
Input: positive integers 𝑥 and 𝑦, each 𝑛 bits longOutput: positive integer 𝑧 where 𝑧 = 𝑥 ⋅ 𝑦
A straightforward approach using base-2 arithmetic, akin to how wemultiply by hand, takes Θ(𝑛2) time.
Could we do better if we used results from subinstances?
![Page 52: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/52.jpg)
Integer Multiplication
Consider the problem of multiplying together two arbitrarily largenumbers.
Input: positive integers 𝑥 and 𝑦, each 𝑛 bits longOutput: positive integer 𝑧 where 𝑧 = 𝑥 ⋅ 𝑦
A straightforward approach using base-2 arithmetic, akin to how wemultiply by hand, takes Θ(𝑛2) time.
Could we do better if we used results from subinstances?
![Page 53: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/53.jpg)
Integer Multiplication
Consider the problem of multiplying together two arbitrarily largenumbers.
Input: positive integers 𝑥 and 𝑦, each 𝑛 bits longOutput: positive integer 𝑧 where 𝑧 = 𝑥 ⋅ 𝑦
A straightforward approach using base-2 arithmetic, akin to how wemultiply by hand, takes Θ(𝑛2) time.
Could we do better if we used results from subinstances?
![Page 54: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/54.jpg)
Integer MultiplicationA Divide and Conquer approach can be considered to be a verylarge scale version of multiplication, only using base 2⌊𝑛
2 ⌋ instead ofa constant base.
For simplicity we assume that 𝑛 is a power of 2, so 𝑛2 will always be
integer.
So we split the bitstring for each of 𝑥 and 𝑦 in half,generating 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 such that
𝑥 = 2𝑛2 ⋅ 𝑥𝐿 + 𝑥𝑅
𝑦 = 2𝑛2 ⋅ 𝑦𝐿 + 𝑦𝑅 .
![Page 55: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/55.jpg)
Integer MultiplicationA Divide and Conquer approach can be considered to be a verylarge scale version of multiplication, only using base 2⌊𝑛
2 ⌋ instead ofa constant base.
For simplicity we assume that 𝑛 is a power of 2, so 𝑛2 will always be
integer.
So we split the bitstring for each of 𝑥 and 𝑦 in half,generating 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 such that
𝑥 = 2𝑛2 ⋅ 𝑥𝐿 + 𝑥𝑅
𝑦 = 2𝑛2 ⋅ 𝑦𝐿 + 𝑦𝑅 .
![Page 56: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/56.jpg)
Integer MultiplicationA Divide and Conquer approach can be considered to be a verylarge scale version of multiplication, only using base 2⌊𝑛
2 ⌋ instead ofa constant base.
For simplicity we assume that 𝑛 is a power of 2, so 𝑛2 will always be
integer.
So we split the bitstring for each of 𝑥 and 𝑦 in half,generating 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 such that
𝑥 = 2𝑛2 ⋅ 𝑥𝐿 + 𝑥𝑅
𝑦 = 2𝑛2 ⋅ 𝑦𝐿 + 𝑦𝑅 .
![Page 57: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/57.jpg)
Using 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 we can now express our multiplication of the𝑛-bit integers as four multiplications of 𝑛
2 -bit integers:
𝑥 ⋅ 𝑦 = (2𝑛2 ⋅ 𝑥𝐿 + 𝑥𝑅) ⋅ (2𝑛
2 ⋅ 𝑦𝐿 + 𝑦𝑅)= 2𝑛 ⋅ 𝑥𝐿𝑦𝐿 + 2𝑛
2 ⋅ (𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿) + 𝑥𝑅𝑦𝑅
Computing this with four half-size multiplications gives us a timerecurrence of
𝑇 (𝑛) = 4𝑇 (𝑛2
) + 𝑐𝑛
![Page 58: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/58.jpg)
Using 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 we can now express our multiplication of the𝑛-bit integers as four multiplications of 𝑛
2 -bit integers:
𝑥 ⋅ 𝑦 = (2𝑛2 ⋅ 𝑥𝐿 + 𝑥𝑅) ⋅ (2𝑛
2 ⋅ 𝑦𝐿 + 𝑦𝑅)= 2𝑛 ⋅ 𝑥𝐿𝑦𝐿 + 2𝑛
2 ⋅ (𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿) + 𝑥𝑅𝑦𝑅
Computing this with four half-size multiplications gives us a timerecurrence of
𝑇 (𝑛) = 4𝑇 (𝑛2
) + 𝑐𝑛
![Page 59: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/59.jpg)
Computing this with four half-size multiplications gives us a timerecurrence of
𝑇 (𝑛) = 4𝑇 (𝑛2
) + 𝑐𝑛
since the operations to split the numbers and put the multiplicationresults together all take time linear in the number of bits:
▶ the numbers are split by bitshifting
▶ combining the recursion results takes three addition operationsand two bitshifts, all linear
![Page 60: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/60.jpg)
Computing this with four half-size multiplications gives us a timerecurrence of
𝑇 (𝑛) = 4𝑇 (𝑛2
) + 𝑐𝑛
since the operations to split the numbers and put the multiplicationresults together all take time linear in the number of bits:
▶ the numbers are split by bitshifting▶ combining the recursion results takes three addition operations
and two bitshifts, all linear
![Page 61: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/61.jpg)
To solve 𝑇 (𝑛) = 4𝑇 (𝑛2 ) + 𝑐𝑛, we can use recursion tree analysis.
Each instantiation makes four calls, each on half the size, and takeslinear time otherwise, so:
𝑇 (𝑛) =log2 𝑛
∑𝑖=0
𝑐 ⋅ 𝑛2𝑖 ⋅ 4𝑖
= 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
4𝑖
2𝑖 = 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
2𝑖
A geometric series
= 𝑐𝑛 ⋅ 2log2 𝑛+1 − 12 − 1
= 2𝑐𝑛 ⋅ 2log2 𝑛 − 𝑐𝑛
= 2𝑐𝑛 ⋅ 𝑛 − 𝑐𝑛 = 2𝑐𝑛2 − 𝑐𝑛 ∈ Θ(𝑛2)
![Page 62: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/62.jpg)
To solve 𝑇 (𝑛) = 4𝑇 (𝑛2 ) + 𝑐𝑛, we can use recursion tree analysis.
Each instantiation makes four calls, each on half the size, and takeslinear time otherwise, so:
𝑇 (𝑛) =log2 𝑛
∑𝑖=0
𝑐 ⋅ 𝑛2𝑖 ⋅ 4𝑖
= 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
4𝑖
2𝑖 = 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
2𝑖
A geometric series
= 𝑐𝑛 ⋅ 2log2 𝑛+1 − 12 − 1
= 2𝑐𝑛 ⋅ 2log2 𝑛 − 𝑐𝑛
= 2𝑐𝑛 ⋅ 𝑛 − 𝑐𝑛 = 2𝑐𝑛2 − 𝑐𝑛 ∈ Θ(𝑛2)
![Page 63: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/63.jpg)
To solve 𝑇 (𝑛) = 4𝑇 (𝑛2 ) + 𝑐𝑛, we can use recursion tree analysis.
Each instantiation makes four calls, each on half the size, and takeslinear time otherwise, so:
𝑇 (𝑛) =log2 𝑛
∑𝑖=0
𝑐 ⋅ 𝑛2𝑖 ⋅ 4𝑖
= 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
4𝑖
2𝑖 = 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
2𝑖
A geometric series
= 𝑐𝑛 ⋅ 2log2 𝑛+1 − 12 − 1
= 2𝑐𝑛 ⋅ 2log2 𝑛 − 𝑐𝑛
= 2𝑐𝑛 ⋅ 𝑛 − 𝑐𝑛 = 2𝑐𝑛2 − 𝑐𝑛 ∈ Θ(𝑛2)
![Page 64: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/64.jpg)
To solve 𝑇 (𝑛) = 4𝑇 (𝑛2 ) + 𝑐𝑛, we can use recursion tree analysis.
Each instantiation makes four calls, each on half the size, and takeslinear time otherwise, so:
𝑇 (𝑛) =log2 𝑛
∑𝑖=0
𝑐 ⋅ 𝑛2𝑖 ⋅ 4𝑖
= 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
4𝑖
2𝑖 = 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
2𝑖
A geometric series
= 𝑐𝑛 ⋅ 2log2 𝑛+1 − 12 − 1
= 2𝑐𝑛 ⋅ 2log2 𝑛 − 𝑐𝑛
= 2𝑐𝑛 ⋅ 𝑛 − 𝑐𝑛 = 2𝑐𝑛2 − 𝑐𝑛 ∈ Θ(𝑛2)
![Page 65: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/65.jpg)
To solve 𝑇 (𝑛) = 4𝑇 (𝑛2 ) + 𝑐𝑛, we can use recursion tree analysis.
Each instantiation makes four calls, each on half the size, and takeslinear time otherwise, so:
𝑇 (𝑛) =log2 𝑛
∑𝑖=0
𝑐 ⋅ 𝑛2𝑖 ⋅ 4𝑖
= 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
4𝑖
2𝑖 = 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
2𝑖
A geometric series
= 𝑐𝑛 ⋅ 2log2 𝑛+1 − 12 − 1
= 2𝑐𝑛 ⋅ 2log2 𝑛 − 𝑐𝑛
= 2𝑐𝑛 ⋅ 𝑛 − 𝑐𝑛 = 2𝑐𝑛2 − 𝑐𝑛 ∈ Θ(𝑛2)
![Page 66: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/66.jpg)
To solve 𝑇 (𝑛) = 4𝑇 (𝑛2 ) + 𝑐𝑛, we can use recursion tree analysis.
Each instantiation makes four calls, each on half the size, and takeslinear time otherwise, so:
𝑇 (𝑛) =log2 𝑛
∑𝑖=0
𝑐 ⋅ 𝑛2𝑖 ⋅ 4𝑖
= 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
4𝑖
2𝑖 = 𝑐𝑛 ⋅log2 𝑛
∑𝑖=0
2𝑖
A geometric series
= 𝑐𝑛 ⋅ 2log2 𝑛+1 − 12 − 1
= 2𝑐𝑛 ⋅ 2log2 𝑛 − 𝑐𝑛
= 2𝑐𝑛 ⋅ 𝑛 − 𝑐𝑛 = 2𝑐𝑛2 − 𝑐𝑛 ∈ Θ(𝑛2)
![Page 67: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/67.jpg)
Gauss’s MethodConsider a different way of computing (𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿), thecoefficient of 2𝑛
2 .
We are already computing 𝑥𝐿𝑦𝐿 and 𝑥𝑅𝑦𝑅
Considering the binomial product
(𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) = 𝑥𝐿𝑦𝐿 + 𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 + 𝑦𝑅𝑥𝑅
we get that
𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 = (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) − 𝑥𝐿𝑦𝐿 − 𝑥𝑅𝑦𝑅
This might be better because we already compute 𝑥𝐿𝑦𝐿 and𝑥𝑅𝑦𝑅
![Page 68: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/68.jpg)
Gauss’s MethodConsider a different way of computing (𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿), thecoefficient of 2𝑛
2 .
We are already computing 𝑥𝐿𝑦𝐿 and 𝑥𝑅𝑦𝑅
Considering the binomial product
(𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) = 𝑥𝐿𝑦𝐿 + 𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 + 𝑦𝑅𝑥𝑅
we get that
𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 = (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) − 𝑥𝐿𝑦𝐿 − 𝑥𝑅𝑦𝑅
This might be better because we already compute 𝑥𝐿𝑦𝐿 and𝑥𝑅𝑦𝑅
![Page 69: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/69.jpg)
Gauss’s MethodConsider a different way of computing (𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿), thecoefficient of 2𝑛
2 .
We are already computing 𝑥𝐿𝑦𝐿 and 𝑥𝑅𝑦𝑅
Considering the binomial product
(𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) = 𝑥𝐿𝑦𝐿 + 𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 + 𝑦𝑅𝑥𝑅
we get that
𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 = (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) − 𝑥𝐿𝑦𝐿 − 𝑥𝑅𝑦𝑅
This might be better because we already compute 𝑥𝐿𝑦𝐿 and𝑥𝑅𝑦𝑅
![Page 70: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/70.jpg)
Gauss’s MethodConsider a different way of computing (𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿), thecoefficient of 2𝑛
2 .
We are already computing 𝑥𝐿𝑦𝐿 and 𝑥𝑅𝑦𝑅
Considering the binomial product
(𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) = 𝑥𝐿𝑦𝐿 + 𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 + 𝑦𝑅𝑥𝑅
we get that
𝑥𝐿𝑦𝑅 + 𝑥𝑅𝑦𝐿 = (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅) − 𝑥𝐿𝑦𝐿 − 𝑥𝑅𝑦𝑅
This might be better because we already compute 𝑥𝐿𝑦𝐿 and𝑥𝑅𝑦𝑅
![Page 71: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/71.jpg)
So the recursive algorithm is:▶ first compute 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 and 𝑥𝐿 + 𝑥𝑅, 𝑦𝐿 + 𝑦𝑅 in linear
time
▶ then calculate 𝑥𝐿𝑦𝐿, 𝑥𝑅𝑦𝑅, and (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅)recursively
▶ and assemble the results in linear timeUsing this approach, we make three recursive calls, each of size 𝑛
2 ,yielding the time recurrence
𝑇 (𝑛) = 3𝑇 (𝑛2
) + 𝑐𝑛
Except that’s not quite right. What we actually have is
𝑇 (𝑛) = 2𝑇 (𝑛2
) + 𝑇 (𝑛2
+ 1) + 𝑂(𝑛)
![Page 72: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/72.jpg)
So the recursive algorithm is:▶ first compute 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 and 𝑥𝐿 + 𝑥𝑅, 𝑦𝐿 + 𝑦𝑅 in linear
time▶ then calculate 𝑥𝐿𝑦𝐿, 𝑥𝑅𝑦𝑅, and (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅)
recursively
▶ and assemble the results in linear timeUsing this approach, we make three recursive calls, each of size 𝑛
2 ,yielding the time recurrence
𝑇 (𝑛) = 3𝑇 (𝑛2
) + 𝑐𝑛
Except that’s not quite right. What we actually have is
𝑇 (𝑛) = 2𝑇 (𝑛2
) + 𝑇 (𝑛2
+ 1) + 𝑂(𝑛)
![Page 73: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/73.jpg)
So the recursive algorithm is:▶ first compute 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 and 𝑥𝐿 + 𝑥𝑅, 𝑦𝐿 + 𝑦𝑅 in linear
time▶ then calculate 𝑥𝐿𝑦𝐿, 𝑥𝑅𝑦𝑅, and (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅)
recursively▶ and assemble the results in linear time
Using this approach, we make three recursive calls, each of size 𝑛2 ,
yielding the time recurrence
𝑇 (𝑛) = 3𝑇 (𝑛2
) + 𝑐𝑛
Except that’s not quite right. What we actually have is
𝑇 (𝑛) = 2𝑇 (𝑛2
) + 𝑇 (𝑛2
+ 1) + 𝑂(𝑛)
![Page 74: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/74.jpg)
So the recursive algorithm is:▶ first compute 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 and 𝑥𝐿 + 𝑥𝑅, 𝑦𝐿 + 𝑦𝑅 in linear
time▶ then calculate 𝑥𝐿𝑦𝐿, 𝑥𝑅𝑦𝑅, and (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅)
recursively▶ and assemble the results in linear time
Using this approach, we make three recursive calls, each of size 𝑛2 ,
yielding the time recurrence
𝑇 (𝑛) = 3𝑇 (𝑛2
) + 𝑐𝑛
Except that’s not quite right. What we actually have is
𝑇 (𝑛) = 2𝑇 (𝑛2
) + 𝑇 (𝑛2
+ 1) + 𝑂(𝑛)
![Page 75: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/75.jpg)
So the recursive algorithm is:▶ first compute 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅 and 𝑥𝐿 + 𝑥𝑅, 𝑦𝐿 + 𝑦𝑅 in linear
time▶ then calculate 𝑥𝐿𝑦𝐿, 𝑥𝑅𝑦𝑅, and (𝑥𝐿 + 𝑥𝑅)(𝑦𝐿 + 𝑦𝑅)
recursively▶ and assemble the results in linear time
Using this approach, we make three recursive calls, each of size 𝑛2 ,
yielding the time recurrence
𝑇 (𝑛) = 3𝑇 (𝑛2
) + 𝑐𝑛
Except that’s not quite right. What we actually have is
𝑇 (𝑛) = 2𝑇 (𝑛2
) + 𝑇 (𝑛2
+ 1) + 𝑂(𝑛)
![Page 76: CS3383 Unit 1: Divide and Conquer Introductionbremner/teaching/cs3383/lectures/01.1-dnc.pdf · ber 7, 2005 Introduction to Algorithms L1.29 and Charles E. Leiserson Merging two sorted](https://reader034.fdocuments.net/reader034/viewer/2022042022/5e792b0f083a0637f408edc2/html5/thumbnails/76.jpg)
Solving the recurrence (board)
𝑇 (𝑛) = 2𝑇 (𝑛2
)+𝑇 (𝑛2
+1)+𝑂(𝑛)
▶ Does the +1 make anydifference? Probably not,but how to be sure?
Figure 2_2 in DPV