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1011 An alternative permutation algorithm is to fill the array a from a [ 0 1 to a [n- , as follows To fill a [ i I generate random num11 bers until you get one that has not been used previously Use an array of Booleans to perform that test Give an analysis of the expected running time (this is tricky) and then write a program that compares this running time with both your analysis and the routine shown in Figure 106

1012 Suppose that you want to generate a random permutation of N distinct items drawn from the range 1 , 2, , M (The case M = N, of course, has already been discussed) Floyd's algorithm does the following First, it recursively generates a permutation of N - 1 distinct items drawn from the range M - 1 It then generates a random integer in the range I to M If the random integer is not already in the permutation we add it; otherwise, we add M a Prove that this algorithm does not add duplicates b Prove that each permutation is equally likely c Give a recursive implementation of the algorithm d Give an iterative implementation of the algorithm

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1013 A random walk in two dimensions is the following game played on the x-J coordinate system Starting at the origin, (0, 0) each iteration consists of a random step either 1 unit left, up, right, or down The walk terminates when the walker returns to the origin The probability of this happening is 1 in two dimensions but less than I in three dimensions) Write a program that performs 100 independent random walks and computes the average number of steps taken in each direction 1014 A simple and effective statistical test is the chi-square test Suppose that you generate N positive numbers that can assume one of M values (for example, we could generate numbers between 1 and M, inclusive) The number of occurrences of each number is a random variable with mean p = N / M For the test to work, you should have y > 10 Let f , be the number of times i is generated Then u compute the chi-square value V = C ( f i ~ ) ~ / The result should be close to M If the result is consistently more than 2& away from M (ie, more than once in 10 tries), then the generator has failed the test Implement the chi-square test and run it on your implementation of the randomInt member function (with low = 1 and h i g h = 100)

see if you can identify any big problems. If the answer is yes, get access to it so that you can make sure all of the settings are set up properly. Specifically, you are going to want to look at the canonicalization settings of the homepage and the filtering of sitelinks. If no one from the organization has yet verified the site through Google Webmaster Tools, then sorting through GWT s diagnostics data can be one of the first valuable services you ll perform for them.

A good discussion of elementary random number generators is provided in [3] The permutation algorithm is due to R Floyd, and is presented in [I] The randomized primality-testing algorithm is taken from [2] and [4] More information on random numbers is available in any good book on statistics or probability 1 J Bentley, "Programming Pearls," Communications of the ACM 30 (1987), 754-757 2 G L Miller, "Riemann's Hypothesis and Tests for Primality," Journal of Computer and System Science 13 (1976), 300-317 3 S K Park and K W Miller, "Random Number Generators: Good Ones Are Hard to Find," Communications of the ACM 31 (1988) 1192-1201 (See also Technical Correspondence in 36 (1993) 105110, which provides the value of A used in Figure 102) 4 M 0 Rabin, "Probabilistic Algorithms for Testing Primality," Journal of Number Theory 12 (1980), 128-138

In this chapter we introduce three important algorithmic techniques and show how to use them by implementing programs to solve two recreational problems The first problem is the word search pu,-zle and involves finding words in a two-dimensional grid of characters The second is optimal play in the game of Tic-Tac-Toe In this chapter, we show: how to use the binary search algorithm to incorporate information from unsuccessful searches and to solve large instances of a word search problem in under 1 sec, how to use the alpha-beta pruning algorithm to speed up the recursive algorithm presented in Section 87, and how to use maps to increase the speed of the tic-tac-toe algorithm

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