Lecture-3-CS210-2012 (1).pptx

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Data Structures and Algorithms

(CS210/ESO207/ESO211)

Lecture 3 Time complexity

Big O notation

Solving a problem: Maximum sum subarray

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Time complexity of an algorithm

Definition:

The time complexity of an algorithm is the worst casenumber of

instructions executed as a functionof the input size(or a

parameter defining the input size).

Example: the time complexity of searching for a 0 in a matrix M[, ] is at

most 2+ cfor some constant c.

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Example:Time complexity of matrix multiplication

Matrix-mult(C[n,n],D[n,n])

{ fori= 0to n-1

{ forj=0to n-1

{ M[i,j]0;

fork=0to n-1

{ M[i,j]M[i,j] + C[i,k]*D[k,j];

}

}

}Return M

}

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Time complexity = + +

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Comparing efficiencyof two algorithms

Let A and B be two algorithms to solve a given problem.

Algorithm A has time complexity : 2 2 + 125

Algorithm B has time complexity : 5 2+ 67 + 400

Question:Which algorithm is more efficient ?

Obviously A is more efficient than B

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Let A and B be two algorithms to solve a given problem.

Algorithm A has time complexity : 2 + 125

Algorithm B has time complexity : 50 + 125

Question:Which one would you prefer based on the efficiencycriteria ?

Answer : A is more efficient thanB for < 50

B is more efficient than A for > 50

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Time complexity is

reallyan issue onlywhen the input is of

large size

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Rule1

We must compare the time complexities of two

algorithms for asymptotically large valueof input size

only

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Algorithm A with time complexity 50 + 125

is certainly more efficient than

Algorithm B has time complexity : 2 + 125

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Exercise 1:

A judgment question for you !

Algorithm A for a given problem has time complexity f()= 5 2 + +1250

Researchers have designed two new algorithms B andC

Algorithm Bhas time complexity g() = 2 + 10

Algorithm Chas time complexity h() = 10 . + 20 + 2000

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Which of B and C is an

improvement over Ain the

true sense ?

lim

g()

f()= 1/5 lim

h()

f()= ?0

C is an improvement over A

in the true sense.

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Another Rule

An algorithm Xis superior to another algorithm Yif

the ratioof time complexity of Xand time complexity

of Yapproaches 0for asymptotically large input size.

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Some insights from the Exercise 1

Algorithm A for a given problem has time complexity f()= 5 2 + +1250

Researchers have designed two new algorithms B andC

Algorithm Bhas time complexity g() = 2 + 10

Algorithm Chas time complexity h() = 10 . + 20 + 2000

Insight 1:multiplicative or additive Constantsdo notplay any role.

Insight 2:the highest order term govern the time complexity asymptotically.

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ORDERNOTATIONS :

A NEATAND PRECISEDESCRIPTION OFTIME COMPLEXITY

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Order notation

Definition: Let f(n)and g(n)be any two increasing functions of n. f(n)issaid to be of the order of g(n) if there exist constants cand such that

f(n) cg(n) for all n>

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f(n)

c g(n)

If f(n)is of the order of g(n),

we write f(n)= O(g(n))

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Order notation :

Examples

20 2= O(2)

100 = O(2)

2 = O(2.)

2000= O(1)

Simple observations:

If f(n)= O(g(n))and g(n)= O(h(n)), then

f(n)= O(h(n))

If f(n)= O(h(n))and g(n)= O(h(n)), then f(n)+g(n)=

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O(h(n))?

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Max-sum subarray problem:

Designing efficient algorithm

Given an array Astoring nnumbers, find its subarraythe sum of

whose elements is maximum?

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3 -5 3 8 2 -4 9 -6 3 -2 -8 3 -5 1 7 -9A

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-218

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A trivial algorithm

A_trivial_algo(A)

{maxA[0];

Fori=0to n-1

Forj=ito n-1

{ tempcompute_sum(A,i,j);

ifmax< temp thenmaxtemp;

}

return max;

}

compute_sum(A, i,j)

{ sumA[i];

Fork=i+1toj sumsum+A[k];

return sum;

}16

Homework: Prove that its

time complexity is O(3)

We shall now design an O() time

algorithm for this problem. You are

advised to make some initial attempts

(few minutes at least). So do not jump

to the next slide.

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Designing an efficient algorithm

Facts from the world of algorithms:

1. There is no formulafor designing efficient algorithms. Almost every

second problem demands a freshapproach.

2. Designing an efficient algorithm or data structure requires1. A deep insightinto the existing efficient algorithms for various problems and

the ability to use that insight at right place.

2. Ability to make key observationabout the problem and its solution.

3. Ability to ask right kind of questionswhen some promising idea fails.

4. A positive attitude and a lot of perseverance.

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We shall demonstrate the above facts

during this course many times.

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Towards designing an O(n) time algorithm

LetS(i): the sum of the largest-sum subarray ending at A[i].

Note that A[i] surely contributes to S(i) (empty subarrays are not considered).

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3 -5 3 8 2 -4 8 -6 3 -2 -8 3 -5 1 7 -9A

i=5

-4

6

-2

9

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S(i)=9for i= 5

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Observations:

In order to solve our problem, it suffices to compute S(i) for all possible 0

i n-1.

Since our objective is to achieve O(n) timecomplexitybound, we need a

way to compute S(i) in O(1) time only. How can we achieve this(ambitious) goal ?

Lesson you should have learnt from the iterative algo for Fibonacci

number

Solve a problem incrementally.

How to use the above insight from Fibonacci number ?

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Question: what is the relation

between S(i) and S(i-1) ?

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IfS(i-1) 0 then

S(i)=A[i]

IfS(i-1) > 0 then

S(i) = S(i-1) + A[i]

Homework:

Prove the above mentioned relation between S(i) and S(i-1).

Design an O(n) time algorithm for Max-sum subarray usingthe above formulation.

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Lecture-3-CS210-2012 (1).pptx

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