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# hw3 - CS 573 Graduate Algorithms Fall 2011 HW 3(due in...

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CS 573: Graduate Algorithms, Fall 2011 HW 3 (due in class on Tuesday, October 18th) This homework contains ﬁve problems. Read the instructions for submitting homework on the course webpage . 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) Given a directed graph G = ( V,E ) and two nodes s,t , an s - t walk is a sequence of nodes s = v 0 ,v 1 ,...,v k = t where ( v i ,v i +1 ) is an edge of G for 0 i < k . Note that a node may be visited multiple times in a walk — this is how it diﬀers from a path. Given G,s,t and an integer k n , design a linear time algorithm to check if there is an s - t walk in G that visits at least k distinct nodes including s and t . Hint: You need to use a linear time algorithm to ﬁnd all strong connected components of a directed graph. Moreover you need to understand the DAG representation of the strong connected components of a graph. You can assume that you have an algorithm for giving you such a representation. Read Chandra’s CS 473 lecture notes if you are unfamiliar with this. 2. (20 pts) You are given a directed graph G = ( V,E ) where each edge e has a length/cost c e (which may be negative) and you want to ﬁnd shortest path distances from a given node s to all the nodes in V . The Bellman-Ford algorithm takes O ( nm ) time where n = | V | and m = | E | while Dijkstra’s algorithm can be implemented in

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hw3 - CS 573 Graduate Algorithms Fall 2011 HW 3(due in...

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