SortingSorting

SortingKeeping data in “order” allows it to be searched more efficiently

Example: Phone Book– Sorted by Last Name (“lots” of work to do this)

• Easy to look someone up if you know their last name• Tedious (but straightforward) to find by First name or

Address

Important if data will be searched many times

Two algorithms for sorting today– Bubble Sort– Merge Sort

Searching: next lecture

Bubble Sort (“Sink” sort here)If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switch

If A(1)>A(2) switch

A(N) is now largest entry

A(N-1) is now 2nd largest entry

A(N) is still largest enry

A(N-2) is now 3rd largest entry

A(N-1) is still 2nd largest entry

A(N) is still largest enry

A(1) is now Nth largest entry.

A(2) is still (N-1)th largest entry.

A(3) is still (N-2)th largest entry.

A(N-3) is still 4th largest entry

A(N-2) is still 3rd largest entry

A(N-1) is still 2nd largest entry

A(N) is still largest entry

Bubble Sort (“Sink” sort here)If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switch

If A(1)>A(2) switch

N-1 steps

N-2 steps

N-3 steps

1 step

22)1( steps of #

21

1

NNNiN

i

Bubble Sort (“Sink” sort here)If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switch

If A(1)>A(2) switch

for lastcompare=N-1:-1:1

for i=1:lastcompare

if A(i)>A(i+1)

Matlab code for Bubble Sortfunction S = bubblesort(A)% Assume A row/column; Copy A to SS = A;N = length(S);for lastcompare=N-1:-1:1 for i=1:lastcompare if S(i)>S(i+1) tmp = S(i); S(i) = S(i+1); S(i+1) = tmp; end endend

What about returning an Index vector Idx, with the property that S = A(Idx)?

Matlab code for Bubble Sortfunction [S,Idx] = bubblesort(A)% Assume A row/column; Copy A to SN = length(A);S = A; Idx = 1:N; % A(Idx) equals Sfor lastcompare=N-1:-1:1 for i=1:lastcompare if S(i)>S(i+1) tmp = S(i); tmpi = Idx(i); S(i) = S(i+1); Idx(i) = Idx(i+1); S(i+1) = tmp; Idx(i+1) = tmpi; end endend

If we switch two entries of S, then exchange the same two entries of Idx. This keeps A(Idx) equaling S

Merging two already sorted arrays

Suppose A and B are two sorted arrays (different lengths)

How do you “merge” these into a sorted array C?

Chalkboard…

Pseudo-code: Merging two already sorted arrays

function C = merge(A,B)nA = length(A); nB = length(B);iA = 1; iB = 1; %smallest unused elementC = zeros(1,nA+nB);for iC=1:nA+nB if A(iA)<B(iB) %compare smallest unused C(iC) = A(iA); iA = iA+1; %use A else C(iC) = B(iB); iB = iB+1; %use B endend

BA nn steps"" of #

MergeSortfunction S = mergeSort(A)n = length(A);if n==1 S = A;else hn = floor(n/2); S1 = mergeSort(A(1:hn)); S2 = mergeSort(A(hn+1:end)); S = merge(S1,S2);end

Base Case

Split in half

Sort 2nd half

Merge 2 sorted arrays

Sort 1st half

Rough Operation Count for MergeSort

Let R(n) denote the number of operations necessary to sort (using mergeSort) an array of length n.

function S = mergeSort(A)n = length(A);if n==1 S = A;else hn = floor(n/2); S1 = mergeSort(A(1:hn)); S2 = mergeSort(A(hn+1:end)); S = merge(S1,S2);end

R(1) = 0

R(n/2) to sort array of length n/2

n steps to merge two sorted arrays of total length n

R(n/2) to sort array of length n/2

Recursive relation: R(1)=0, R(n) = 2*R(n/2) + n

Rough Operation Count for MergeSortThe recursive relation for R R(1)=0, R(n) = 2*R(n/2) + nClaim: For n=2m, it is true that R(n) ≤ n log2(n)

Case (m=0): true, since log2(1)=0

Case (m=k+1 from m=k)

12 1 kk

kkk 222log22 2 kkR 2222 kk RR 222 1

12

1 2log2 kk

Recursive relation

Induction hypothesis

Matlab command: sortSyntax is [S] = sort(A)If A is a vector, then S is a vector in ascending

order

The indices which rearrange A into S are also available.

[S,Idx] = sort(A)

S is the sorted values of A, and A(Idx) equals S.