Running Time Calculations in Algorithm Analysis, Slides of Data Structures and Algorithms

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7/7/2012 Algorithm Analysis 2-

Chapter 2

Algorithm Analysis

EE 232 Data Structures Session-05 , Spring-

7/7/2012 Algorithm Analysis 2-

Chapter 2: Algorithm Analysis

2.1 Mathematical Background

2.2 Model

2.3 What to Analyze

2.4 Running Time Calculations

7/7/2012 Algorithm Analysis 2-

Lets Recall

 What is an Algorithm?  a clearly specified set of simple instructions to be followed to solve a problem

• Takes a set of values, as input and
• produces a value, or set of values, as output

 What are Data structures?  Methods of organizing data

 Program = algorithms + data structures

7/7/2012 Algorithm Analysis 2-

Lets Recall…

 Why should we analyse an algorithm?

 Writing a working program is not good enough!  The program may be inefficient  If the program is run on a large data set, then the running time becomes an issue

• Recall the selection problem

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Mathematical Background

 We need a theoretical framework upon which we can compare algorithms

 The idea is to establish a relative order among different algorithms, in terms of their relative rates of growth (of the running time)

 The rates of growth are expressed as functions, which are generally in terms of the number of inputs n e.g. a running time function can be expressed as T( n )

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Mathematical Background…

 Such functions describe the utilization of some precious resource (e.g. time or space) as the size of the problem increases

 We can determine the exact running time of an algorithm, however for large inputs, the multiplicative constants and lower-order terms are dominated by the effects of the input size itself

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Mathematical Background…

 *If we look at a slightly larger range from n = [0, 10], we begin to note that they are more similar:

*http://www.ece.uwaterloo.ca/~ece250/ docsity.com

7/7/2012 Algorithm Analysis 2-

Mathematical Background…

 *Extending the range to n = [0, 100], the similarity increases:

*http://www.ece.uwaterloo.ca/~ece250/ docsity.com

7/7/2012 Algorithm Analysis 2-

Mathematical Background…

 *They are different absolutely, for example, f(1000) = 1 000 000 g(1000) = 997 002  however, the relative difference is very small

 and this difference goes to zero as n → ∞  The justification for the pair of polynomials being similar is that, both have the same leading term:  n^2

f( 1000 )

f( 1000 ) g( 1000 )

*http://www.ece.uwaterloo.ca/~ece250/ docsity.com

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Mathematical Background…

We can throw away the lower-order terms and ignore the leading coefficients of the highest term as the problem size grows really, really large

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Mathematical Background…

Asymptotic Notation  The “big-Oh” O - Notation  f( n ) 1 = O (g( n )) , if there exists positive constants c and n 0 , such that f( n )  cg( n ) for all n ≥ n 0

 f( n ) and g( n ) are functions over non-negative integers  The growth rate of f( n ) is less than or equal to that of g( n )

f ( n)

c g n  ( )

n (^0) Input Size

Running Time

1: means set membership i.e. f( n ) Є O (g( n ))

7/7/2012 Algorithm Analysis 2-

Mathematical Background…

 g( n ) is an upper bound on f( n )  O -notation provides an asymptotic upper bound on a function  Used for worst-case analysis

Examples  If f( n ) = 2 n^2 , then f( n ) = O ( n^4 ) , f( n ) = O ( n^3 ) and f( n ) = O ( n^2 ) are all technically correct but the last answer is the best

f( n)

c g n  ( )

n 0 Input Size

Running Time

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Mathematical Background…

 W–notation provides an asymptotic lower bound on a function  Used to describe best- case running times

Examples  n^3 grows faster than n^2 , so we can say that n^3 = W( n^2 )

Input Size

Running Time

f ( n) c g n  ( )

n 0

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Mathematical Background…

 The “big-Theta” Q-Notation

 f( n ) = Q(g( n )) if there exists positive constants c 1 , c 2 , and n 0 , such that c 1 g( n )  f( n )  c 2 g( n ) for all n  n 0

 for all values of n to the right of n 0 , the values of f( n ) lies at or above c 1 g( n ) and at or below c 2 g( n )

Input Size

Running Time

f ( n)

n 0

c 2 g( n)

c 1 g( n)