Ch. 10 Measuring the Efficiency of Algorithms

Three difficulties with comparing programs instead of algorithms

How are the algorithms coded? What computer should you use? What data should the programs use?

Average-case analysis

A determination of the average amount of time that an algorithm requires to solve problems of size n

Worst-case analysis

A determination of the maximum amount of time that an algorithm requires to solve problems of size n

Growth-rate function

A mathematical function used to specify an algorithm's order in terms of the size of the problem

Big O notation

A notation that uses the capital letter O to specify an algorithm's order Example: O(f(n))

Definition of the order of an algorithm

Algorithm A is order f(n) - denoted O(f(n)) - if constants k and n0 exist such that A requires no more than k * f(n) time units to solve a problem of size n ≥ n0

Counting an algorithm's operations is a way to access its efficiency

An algorithm's execution time is related to the number of operations it requires

An algorithm's growth rate

Enables the comparison of one algorithm with another

Order of growth of some common functions

O(1) < O(log2n) < O(n) < O(n * log2n) < O(n2) < O(n3) < O(2n)

Analysis of algorithms

Provides tools for contrasting the efficiency of different methods of solution

A comparison of algorithms

Should focus of significant differences in efficiency Should not consider reductions in computing costs due to clever coding tricks

Algorithm analysis should be independent of

Specific implementations Computers Data

Properties of growth-rate functions

You can ignore low-order terms You can ignore a multiplicative constant in the high-order term O(f(n)) + O(g(n)) = O(f(n) + g(n))

Algorithm efficiency is typically a concern for

large problems only

An algorithm's time requirements

requirements can be measured as a function of the problem size