4 5 1 return - expectedvalue

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* log( randomReal(

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Generation of a random number according to the negative exponential distribution

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Another important nonuniform distribution is the negative exponential distribution, shown in Figure 105, which has the same mean and variance and is used to model the time between occurrences of random events We use it in the simulation application shown in Section 142 Many other distributions are commonly used Our main purpose here is to show that most can be generated from the uniform distribution Consult any book on probability and statistics to find out more about these functions

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The negative

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distribution has the a,d variance It is used to model the time between occurrences of random events

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104 Generating a Random Permutation

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Consider the problem of simulating a card game The deck consists of 52 distinct cards, and in the course of a deal, we must generate cards from the deck, without duplicates In effect, we need to shuffle the cards and then iterate through the deck We want the shuffle to be fair That is, each of the 52! possible orderings of the deck should be equally likely as a result of the shuffle

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1 / / Generate a random permutatior of lr 2 void permuze( vectoriint> & a )

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3 4 5

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Random r( (int) time( 0 ) const int n = asize ( ) ; for( int i = 0; i < n; i++ a[ i ] = i + 1 ;

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State set by time

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for( int j = 1; j < n; jt+ ) swap( a[ j 1 , a[ rrandomInt( 0 , j i I

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A permutation routine

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A random permutation can be generated in linear time, using one random number per item

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The correctness of permute i s

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This type of problem involves the use of a random permutation In general the problem is to generate a random permutation of 1 , 2, , N, with all permutations being equally likely The randomness of the random permutation is, of course, limited by the randomness of the pseudorandom number generator Thus all permutations being equally likely is contingent on all the random numbers being uniformly distributed and independent We demonstrate that random permutations can be generated in linear time, using one random number per item A routine, permute to generate a random permutation is shown in Figure 106 In the permute routine the first loop initializes the permutation with 1, 2, , N The second loop performs a random shuffling In each iteration of the loop we swap a [ j 1 with some array element in positions 0 to j (it is possible to perform no swap) Clearly, permute generates shuffled permutations But are all permutations equally likely The answer is both yes and no The answer based on the algorithm is yes There are N! possible permutations, and the number of different possible outcomes of the h'- 1 calls to randomInt at line 11 is also N! The reason is that the first call produces 0 or 1 so it has two outcomes The second call produces 0 1, or 2, so it has three outcomes The last call has N outcomes The total number of outcomes is the product of all these possibilities because each random number is independent of the previous random numbers All we have to show is that each sequence of random numbers corresponds to only one permutation We can do so by working backward (see Exercise 106) However the answer is actually no-all permutations are not equally likely There are only 2" - 2 initial states for the random number generator,

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Randomized Algorithms

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so there can be only 3 1- 2 different permutations This condition could be a problem in some situations For instance, a program that generates 1,000,000 permutations (perhaps by splitting the work among many computers) to measure the performance of a sorting algorithm almost certainly generates some permutations twice-unfortunately Better random number generators are needed to help the practice meet the theory Note that rewriting the call to s w a p with the call to r randomInt ( 0, 1) does not work, even for three elements There are 3! = 6 possible npermutations, and the number of different sequences that could be computed by the three calls to randomInt is 3 3 = 27 Because 6 does not divide 27 exactly, some permutations must be more likely than others

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