hw4 - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 4 Due at the beginning of class on Wednesday, October 17, 2007 Problem 1 (10 points) We analyzed the process of throwing n balls into n bins independently and at random and showed that the maximum load is at most O ( log n log log n ) with high probability. Suppose instead that the balls are assigned to bins by a function f which is chosen from a universal family. Prove that, with probability at least 1 / 2, the maximum load is O ( n ). Hint: Let X i be the load of the i th bin. Give an upper bound on E [ X 2 i ]. Problem 2 (15 points) In order to implement a counter capable of counting from 1 to N, at least log N bits are needed. In some cases, allowing this number of bits is too expensive. If we allow the counter to make some errors, we can get by with much fewer bits. In this question you’ll see how to implement a “probabilistic counter” - a counter that counts with small errors, and uses only Θ(log log N ) bits. The counter is defined as follows: Initially the counter is 0. An Increment operation is performed as follows: If the current value of the counter is i , increase the counter value by 1 with probability 1 2 i , and leave the counter in its
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This homework help was uploaded on 01/29/2008 for the course COS 340 taught by Professor Charikarandchazelle during the Fall '07 term at Princeton.

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hw4 - Computer Science 340 Reasoning about Computation...

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