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Presentation transcript:

Andrey Markov, public domain image Numerical Algorithms 10/12/2017 5:24 PM Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 Chernoff Bounds Andrey Markov, public domain image © 2015 Goodrich and Tamassia Chernoff Bounds

Markov’s Inequality © 2015 Goodrich and Tamassia Chernoff Bounds

Sums of Indicator Random Variables Let X = X1 + X2 + . . . + Xn be the sum of n independent 0-1 indicator random variables, such that Xi = 1 with probability pi. Using the language of probability theory, X is a random variable from the binomial distribution. Intuitively, X is equal to the number of heads one gets by flipping n coins such that the i-th coin comes up heads with probability pi. Define the mean, or expected value of X, as follows: © 2015 Goodrich and Tamassia Chernoff Bounds

A Chernoff (upper) Bound © 2015 Goodrich and Tamassia Chernoff Bounds

A Chernoff (lower) Bound © 2015 Goodrich and Tamassia Chernoff Bounds

Application: Load Balancing Suppose we have a set of n processors and a set of n jobs for them to perform, but no good way of assigning jobs to processors. So we just assign jobs to processors at random. What is a good high-probability upper bound on the number of jobs assigned to any processor? We can answer this question using a Chernoff bound. Let X be a random variable representing the number of jobs assigned to processor 1. Then X can be written as X = X1 + X2 + . . .Xn, where Xi is the 0-1 indicator random variable that job i is assigned to processor 1. Thus, Pr(Xi = 1) = 1/n and μ = E[X] = 1. © 2015 Goodrich and Tamassia Chernoff Bounds

Applying a Chernoff Bound Since the Xi’s are clearly independent, we can apply the Chernoff bound from Theorem 19.12 to get, for any integer m > 1: Thus, the probability that processor 1 has more than m jobs assigned to it by this random assignment is at most 1/n2. Therefore, the probability that any processor is assigned more than mjobs is at most n/n2 = 1/n. In other words, the number of processors assigned to any processor is O(log n/ log log n) with high probability. © 2015 Goodrich and Tamassia Chernoff Bounds