ECON
Summer 2011 Final

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

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• dang026tangwenye11
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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
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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• Fall '11
• Arthur
• sh, ar stu, ed d, NP-complete problems, Travelling salesman problem, polynomial time

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