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practice1_solution - CS648 Randomized Algorithms Semester...

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CS648 Randomized Algorithms Semester II, 2007-08 Motivational problem sheet Note : These problems are for those students whose expectation from this course is at least little more than just getting a good grade. More and more problems will be added to this sheet. So keep on downloading it every week at least. 1. ( * * * ) Till now we always proved low probabilty bound on positive deviation of a random variable from its expected value. This time we shall work in the other direction as well. Here is an example. We showed in an earlier class that if we throw n balls independently uniformly into n bins, the maximum load is O (log n ) with high probability. In fact, the equation ( x c ) x = n 3 we solved has exact solution x = b log n log log n for some constant b . You have to prove that this bound is tight from below too. That is, show that with high probability the maximum load is Ω( log n log log n ) . 2. ( ** ) (Due to T. Cover and M. Rabin) Consider the following game. A friend writes down two numbers on two slips of paper and then randomly puts one in one hand and the other in the other hand. You get to
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