hw1 - CSci 5403, Spring 2010 Homework 1 due: Feb 2, 2010 1....

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Unformatted text preview: CSci 5403, Spring 2010 Homework 1 due: Feb 2, 2010 1. Constructibility. Recall that a function t : N N is time-constructible if there is a Turing Machine that computes the binary representation of t ( n ) given 1 n as input, in time O ( t ( n )). Suppose f and g are time-constructible, and prove that the following are also time-constructible: (a) f ( n ) + g ( n ) (b) f ( n ) g ( n ) (c) f ( g ( n )) (d) 2 f ( n ) 2. Approximate Cuts. The max-cut problem is defined as follows. Given an undirected graph G = ( V,E ), a cut is a set of vertices S V . The weight of the cut is the number of edges that cross between S and V \ S , i.e. wt G ( S ) = | E S ( V \ S ) | . The decision problem max-cut is defined by {h G,k i | G = ( V,E ) S V.wt G ( S ) k } , and the search problem is to find a cut with weight max( G ) = max S V wt G ( S ). It is possible to show that max-cut is NP-hard. This question will explore randomized algorithms for the search problem. A randomized algorithm is one that can make indepen- dent, unbiased coin flips. The output of the algorithm on any given input is thus a random variable, and we can perform the usual operations on random variables (computing func-...
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hw1 - CSci 5403, Spring 2010 Homework 1 due: Feb 2, 2010 1....

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