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Algorithms

Algorithms. Randomized Algorithms Major Sources: Brassard and Bratley, Algorithmics: Theory and Practice Cormen, Lieserson, Rivest, and Stein, Introduction to Algorithms Horowitz, Sahni, and Rajasekaran, Computer Algorithms in C++. Randomized Algorithms.

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Algorithms

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  1. Algorithms Randomized Algorithms Major Sources: Brassard and Bratley, Algorithmics: Theory and Practice Cormen, Lieserson, Rivest, and Stein, Introduction to Algorithms Horowitz, Sahni, and Rajasekaran, Computer Algorithms in C++

  2. Randomized Algorithms • Randomized algorithms make use of a randomizer (such as a random number generator). • Some of the decisions of the algorithm depend on the output of the randomizer. • The output may vary from run to run. • The execution time may vary from run to run.

  3. Major Classes of Randomized Algorithms • Las Vegas algorithms: • Always produce the same (correct) output for the same input • Execution time depends on the randomizer • Goal is often to minimize the odds of encountering worst case performance • Example: Randomized quick sort • Monte Carlo algorithms: • Occasionally gives an incorrect answer • The probability of an incorrect answer is required to be low. • Typically does not display variation in execution time for a particular input • Goal is to provide an answer that has a high likelihood of being correct in a reasonable time • Example: Determining primality of an integer with several hundred decimal digits.

  4. Analysis of Probabilistic Algorithms • Analysis is often complex and typically requires concepts from • Probability • Statistics • Number theory • Assume availability of Random(a,b) which returns an integer between a and b, inclusive with each integer being equally likely. • Assume cost of producing a single random value is constant • In practice we typically use pseudo-random number generators

  5. Randomly Permuting Arrays • Many randomized algorithms randomize the input by randomly permuting the elements of an array. • Two algorithms given in text • Permute by sorting (page 101) • Randomize in place (page 103)

  6. RANDOMIZE-IN-PLACE(A) • n  length[A] • for i  1 to n do • swap A[i]  A[RANDOM(i,n)] • How many permutations are there of n items? • For each permutation of n items, what is the probability that permutation will be produced by the above algorithm?

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