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Unformatted text preview: CS 573: Graduate Algorithms, Fall 2011 HW 4 (due in class on Tuesday, November 1st) This homework contains five problems. Read the instructions for submitting homework on the course web page . In particular, make sure that you write the solutions for the problems on separate sheets of paper. Write your name and netid on each sheet. Collaboration Policy: For this home work students can work in groups of up to three students each. Only one copy of the homework is to be submitted for each group. Make sure to list all the names/netids clearly on each page. Note on Proofs: Details are important in proofs but so is conciseness. Striking a good balance between them is a skill that is very useful to develop, especially at the graduate level. 1. (20 pts) Sampling is a powerful methodology in randomized algorithm design although we did not get to see it formally yet. The idea is that a small random sample of the input retains many interesting properties of the original input with good probability; one can then run an algorithm on the sample instead of the original input. This problem gives an example. Let S be a set of n numbers (assume they are distinct) and say you want to approximate their median. A number x is an-approximate median of S if at least ( 1 2- ) n numbers in S are less than x and at least ( 1 2- ) n are greater than x . Consider the following algorithm. Pick a subset S S uniformly at random and return the median of the sample S as the approximate median. Show that there is a constant c independent of n such that if the sample size | S | c then the output is a 0 . 05-approximate median with probability at least 0 . 99. The99....
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