CMPSCI 711: Really Advanced Algorithms Micah Adler Summary: CMPSCI 711: Really Advanced Algorithms Micah Adler Problem Set 1 Out: February 11, 2003 Due: February 20, 2003 1. You are given a coin whose probability of heads is an unknown value p such that 0 < p < 1. Devise a method of using this coin to produce a sequence of completely independent bits, each of which is equally likely to be 0 or 1. Your scheme should have the property that the expected number of coin tosses per bit generated is at most 1 p(1 p) . 2. Suppose dn ln ne balls are thrown independently into n urns, where each ball has a 1=n chance of landing in any given urn. Show that there is a constant c such that, with probability at least 1 1=n, no urn receives more than c ln n balls. 3. Say we have a decision problem , and a randomized algorithm A for , such that on any given input, A may return the wrong answer, but Pr[A is wrong]  1=2 1=p(n), for some polynomially bounded function p(n) of the input size n. Use a Cherno bound to show that a polynomial number of repetitions of A can be used to reduce the error probability to 1=2 n . 4. Consider the following variant of the left-to-right routing scheme for the hypercube: for each packet, we randomly order the n bit positions of the node labels and then correct the mismatched bits between the source and the destination of the packet in that order. Show that there is a permutation for which, Collections: Computer Technologies and Information Sciences