Unit 2 - Analysis of Algorithm

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Analyzing the Algorithm

Chapter 1


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Introduction An algorithm is an effective method for solving a

problem expressed as a finite sequence ofinstructions.

In world of computer science the word algorithm has

a great significant.

Algorithm is developed first , before writing the

program.

Efficiency of the program is improved by using Good

Algorithm.


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Algorithm

An algorithm is a finite set of instructions that if

followed , accomplishes a particular task in a finite

amount of time. Every algorithm must satisfy the following criteria.

1)Input

2)Output

3)Definiteness

4)Finiteness

5)Effectiveness


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Characteristics

1) Input: Zero or more quantities are externally

supplied

2) Output: At least one quantity is produced3) Definiteness: Each instruction is clear and

unambiguous

4) Finiteness: The algorithm terminates in a finitenumber of steps.

5) Effectiveness: Every step in the algorithm should be

easy to understand & can be implemented using any

programming language.


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Analysis Of algorithm Analysis is the process of breaking a complex topic

into a smaller parts to gain a better understanding ofit.

To analyze an algorithm is to determine the amount of

resources (such as time and storage) necessary toexecute it.

Analysis of algorithm means developing a prediction

of how fast the algorithm works based on the problemsize.


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Algorithms Performance An algorithms performance depends on

1) Internal factors

2) External factors

External Factors:-

a) Speed of the computer on which it is run

b) Quality of the compiler

Internal Factors:-

a) Time Required to run

b) Space


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Performance of Program

To develop a program, we need two major factors:

Algorithm

Data Structure

The choice of good data structures and algorithm design

methods impacts theperformance of programs.

The performance of a program can be analyzed by its

complexity i.e. the amount of computer memory and time

needed to run a program.

Space Complexity

Time Com lexit


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Complexity Computational complexity is a characterization of the

time or space requirements for solving a problem by aparticular algorithm.

The analysis of the program requires two main

considerations.1) Time complexity.

2)Space complexity.


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Time Complexity The time complexity of a program/algorithm is the

amount o computer time that it needs to run to

completion. OR The total amount of time taken by a program for

execution is the time complexity of a program.

For measuring the time complexity of an algorithm,we concentrate on developing only the frequencycount for all key statements (i.e statements that are

important & are the basic instructions of analgorithm.)

Complexity of any program is represented by Big Onotation.


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Time Complexity contd..

Frequency count is defined as the total number oftimes statements are executed.

Thus, Execution time is proportional to the frequencycount of an active operation.

Eg. Algorithm A contains a= a+1

The statement a=a+1 is independent. The number oftimes this shall be executed is 1. Therefore frequencycount of algorithm A is 1.

Eg. 2) for(j=1; j


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Time Complexity contd..

Amount of Computer time required by a program to execute

Linear Complexity O (n)

Quadratic Complexity O (n2)

Cubic Complexity O (n3)

Logarithmic Complexity O (log n)

Exponential Complexity O (2n)


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Space Complexity The space complexity of an algorithm or program is

the amount of memory that it needs to run tocompletion.

The space needed by the algorithm is the sum of the

following components.1) Fixed Part

2) Variable space requirement


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Space Complexity contd

Fixed Part:- Fixed part is not dependent on the characteristics of the

inputs and outputs.

Fixed part consist of space for fixed size structure

variables & constants.

Variable Space Requirement:- This includes a space, needed by variables whose size

depends upon the particular problem being solved and

the stack space required for recursion.


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for(int i= 0; i< n; i++)

{

DoProcessing();for(int j= i; j< n; j++)

{

EnterStuffinDBMS();

for(int p=1; p< n; p++ )

{

QuerySomeResults();}

for(int l= 0; l< m; l++)

{

}

}for(int k= 2; k< m; k++)

{

}

}

i

j

p

l

k

To Calculate the Time Complexity


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O ( n * ( n * ( n + m ) + m ) )

O ( n * ( n2 + nm + m))

O (n3 + n2m + nm)

O (n3) Time Complexity of the program

To Calculate the Time Complexity (continued)

i

j

p

l

k


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Space Complexity V/s Time Complexity

A software which executes in less time requires more space

and vice versa.

This is known as tradeoff between Space and Time

complexity.

T (Time)

S (Space)


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Best Case ,Worst Case & Average Case

In some cases, it is important to consider the best,

worst and/or average (or typical) performance of an

algorithm.

There are different types of time complexities which

can be analyzed for an algorithm.

1) Best Case Time Complexity

2) Average Case Time Complexity3) Worst case Time Complexity.


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Best Case Time Complexity

It is a measure of the minimum time that thealgorithm will require for an input of size n.

If an input data of n items is presented in sorted

order ,the operations performed by the algorithm willtake the least time.


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Worst Case Time Complexity

It is a measure of the maximum time that thealgorithm will require for an input of size n.

If n input data items are supplied in reverse orderfor any sorting algorithm, then the algorithm will

require n^2 operations to perform the sort which will

correspond to the worst case time complexity of thealgorithm.


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Average Case Time Complexity

The average time that an algorithm will require toexecute a typical data of size n is known as average

case time complexity.

The value that is obtained by averaging the running

time of an algorithm for all possible inputs of size n

can determine average case time complexity.


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Example For example, consider the deterministic sorting

algorithm quicksort .

This solves the problem of sorting a list of integerswhich is given as the input.

The best-case scenario is when the input is already

sorted, and the algorithm takes time O(n log n) for suchinputs.

The worst-case is when the input is sorted in reverse

order, and the algorithm takes time O(n2

) for this case. If we assume that all possible permutations of the input

list are equally likely, the average time taken for sorting

is O(n log n).


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Asymptotic Notation Running time of an algorithm as a function

of input size n (for large n). Expressed using only the highest-order term in the

expression for the exact running time.

f(n)=1+n+n^2

Order of polynomial is the degree of the highest term.

O(f(n))=O(n^2)


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Asymptotic Notation1) Big-Oh

2) Omega

3)Theta


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Big-oh Notation The time complexity of a algorithm can be expressed

in terms of the order of magnitude of frequency countusing the Big-Oh notation

When we have only an asymptotic upper bound , we

use O-notation. O(g(n)) = { f(n) :there exist positive constants c & n0

such that

0


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Big-oh Notation


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Theta Notation () The theta notation asymptotically bounds a function

from above & below. A function f(n) belongs to the set O(g(n)) if there

exist positive constants c1 & c2 such that it can be

sandwiched between c1g(n) & c2g(n) , forsufficiently large n.

(g(n)) = {f(n) : there exist positive constants c1,

c2,& n0 such that0 c1g(n) f(n) c2g(n) for all n>=n0 }


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Theta Notation


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Omega Notation Omega notation provides an asymptotic lower

bound. For functiong(n), we define (g(n)),Omega of

n, as the set:

(g(n)) = {f(n) :there exist positive constantsc & n0 such that

0 cg(n) f(n) for all n n0}


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Omega Notation


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Complexity Of Algorithms Measure the time complexity (in terms of

steps) as a function of the input size. f(n)=12n3+1

Unit 2 - Analysis of Algorithm

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