# hbs13 - CS 473 Algorithms Fall 2010 HBS 13 Final Review...

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CS 473: Algorithms, Fall 2010 HBS 13: Final Review Problem 1 . [Recurrences] Solve the following recurrences: T ( n ) = T ( b n/ 15 c ) + T ( b n/ 10 c ) + 2 T ( b n/ 6 c ) + n T ( n ) = T ( n - 1) + 2 n - 1, where T (0) = 0 Problem 2 . [Graphs (3.10)] Let G = ( V,E ) be an unweighted, undirected graph and let u and v be two vertices of G . Describe a linear time algorithm to ﬁnd the number of shortest paths from u to v . Note that we only want the number of paths as there may be an exponential number of them. Give an example graph with an exponential number of ( u,v )-paths. Problem 3 . [Shortest Paths Reduction] Consider a system of m linear inequalities over n variables { x 1 , ··· ,x n } . The k -th inequality is in the form x k 1 - x k 2 t k for 1 k 1 ,k 2 n and constant t k (could be positive, zero or negative). The task is to present an algorithm that ﬁnds a solution of this system or indicates that the system has no solution. Build a graph G on vertex set { v 1 , ··· ,v n ,s } . Put a directed edge from s to every v i with weight 0. If the k -th inequality is x k 1 - x k 2 t k , then put a directed edge from v k 2 to v k 1 in the graph with weight t k . Let d ( s,v i ) be the length of the shortest path from s to v i in the graph G . Are the values d ( s,v i ) guaranteed to exist? What algorithm could we use to compute all the values d ( s,v i ) if they exist? Assuming that all

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## This note was uploaded on 01/22/2011 for the course CS 473 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.

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hbs13 - CS 473 Algorithms Fall 2010 HBS 13 Final Review...

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