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mar5 - Illinois Institute of Technology Department of...

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Illinois Institute of Technology Department of Computer Science Lecture 12: March 5 CS 330 Discrete Mathematics Spring Semester, 2008 1 Analysis of algorithms In this section, we will return to the problem that originally led us into the discussion of probability: For example, consider the algorithm to find the maximum of n numbers x 1 , x 2 , . . . , x n listed below. To analyze this algorithm fully we want to know exactly how many times each of the instructions is executed; but for the moment, let us ask only how many times the statement “ m k ” is executed. The answer, of course, depends not on the particular values of the x i , but rather on their relative order: If x 1 is the maximum of the n elements, the statement will never be executed. If x 1 < x 2 < · · · < x n then the statement will be executed n - 1 times. For how many of the relative arrangements of the inputs x 1 , x 2 , . . . , x n will the statement be executed exactly i times? The answer to this question is at the heart of the analysis of the algorithm. ALGORITHM 1: /* Selecting the largest of x [1] , x [2] , . . . , x [ n ] . */ m 1 FOR k 2 TO n DO IF x k > x m THEN m k | {z } “assignment” of interest
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