Summer 2011 Final - CS570 Analysis of Algorithms Summer...

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CS570 Analysis of Algorithms Summer 2011 Final Exam Name: _____________________ Student ID: _________________ ____Check if DEN student Maximum Received Problem 1 20 Problem 2 13 Problem 3 14 Problem 4 13 Problem 5 20 Problem 6 20 Total 100 This study resource was shared via CourseHero.com
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2 hr exam Close book and notes 1) 20 pts Mark the following statements as TRUE or FALSE . No need to provide any justification except for the question at the bottom of the page. [ TRUE/ FALSE ] In a flow network, if all edge capacities are distinct, then the max flow of this network is unique. [ TRUE /FALSE ] To find the minimum element in a max heap of n elements, it takes O(n) time. [ TRUE /FALSE ] Let T be a spanning tree of graph G(V, E), let k be the number of edges in T, then k=O(V) [ TRUE /FALSE ] Linear programming problems can be solved in polynomial time. [ TRUE /FALSE ] Consider problem A: given a flow network, find the maximum flow from a node s to a node t. problem A is in NP. [ TRUE /FALSE ] Given n numbers, it takes O(n) time to construct a binary min heap. [ TRUE /FALSE ] Kruskal's algorithm for finding the MST works with positive and negative edge weights. [ TRUE/ FALSE ] Breadth first search is an example of a divide-and-conquer algorithm. [ TRUE/ FALSE ] If a problem is not in P, then it must be in NP. [ TRUE/ FALSE ] L1 can be reduced to L2 in Polynomial time and L1 is in NP, then L2 is in NP This study resource was shared via CourseHero.com
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2) 13 pts Imagine that you constructed an approximation algorithm for the Traveling Salesman Problem that could always calculate a solution that is correct within a factor of 1/ k of the optimal tour in O( n 2k ) time. Would you be able to use this approximation algorithm to obtain a “good” solution to all other NP-Complete problems? Explain why or why not. Yes. You can use it. In the Traveling Salesperson Problem, we are given an undirected graph G = (V,E) and cost c(e) > 0 for each edge e 2 E. Our goal is to find a Hamiltonian cycle with minimum cost. A cycle is said to be Hamiltonian if it visits every vertex in V exactly once. TSP is known to be NP-complete, and so we cannot expect to exactly solve TSP in polynomial time. What is worse, there is no good approximation algorithm for TSP unless P = NP. This is because if one can give a good approximation solution to TSP in polynomial time, then we can exactly solve the NP-Complete Hamiltonian cycle problem (HAM) in polynomial time, which is impossible unless P = NP. Recall that HAM is the
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