CS70%20Midterm_1_Fall%202008-%20Solution

# CS70%20Midterm_1_Fall%202008-%20Solution - CS570 Analysis...

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Unformatted text preview: CS570 Analysis of Algorithms Fall 2008 Exam I Name: _____________________ Student ID: _________________ ____Monday Section ____Wednesday Section ____Friday Section Maximum Received Problem 1 20 Problem 2 10 Problem 3 10 Problem 4 20 Problem 5 20 Problem 6 20 Total 100 2 hr exam Close book and notes 1) 20 pts Mark the following statements as TRUE or FALSE . No need to provide any justification. [ TRUE/FALSE ] F Given graph G and a Minimum Spanning Tree T on G , you could find the (weighted) shortest path between arbitrary pair u, v in V(G) using only edges in T . [ TRUE/FALSE ] F V 2 log V = θ (E logE 2 ) whether the graph is dense or sparse. [ TRUE/FALSE ] T If DFS and BFS returns different trees, then the original graph is not a tree. [ TRUE/FALSE ] T An algorithm with the running time of n * 2 min(n log n, 10000) runs in polynomial time. [ TRUE/FALSE ] T We may need to run Dijkstra’s algorithm to compute the shortest path on a directed graph, even if the graph doesn't have a cycle. [ TRUE/FALSE ] F Given a graph that contains negative edge weights, we can use Dijkstra's algorithm to find the shortest paths between any two vertexes by first adding a constant weight to all of the edges to eliminate the negative weights. [ TRUE/FALSE ] F If f(n) and g(n) are asymptotically positive functions, then f(n)+g(n) = θ( min{f(n),g(n)}). [ TRUE/FALSE ] F For a stable matching problem such that m ranks w last and w ranks m last, m and w will never be paired in a stable matching. [ TRUE/FALSE ] F Breadth first search is an example of a divide-and-conquer algorithm. [ TRUE/FALSE ] T Kruskal's algorithm for finding the MST works with positive and negative edge weights. 2) 10 pts a) Arrange the following in the increasing order of asymptotic growth. Identify any ties. lg n 10 , 3 n , lg n 2n , 3n 2 , lg n lg n , 10 lg n , n lg n , n lg n Solution: The order is lg n 10 , lg n lg n , lg n 2n , n lg n , 3n 2 , 10 lg n , n lg n , 3 n Where lg n 2n and n lg n ties b) Analyze the complexity of the following loops: i- x = 0 for i=1 to n x= x + lg n end for O(n) ii- x=0 for i=1 to n for j=1 to lg n x = x * lg n endfor endfor O(nlgn) iii- x = 0 k = “some constant” for i=1 to max (n, k) x= x + lg n end for O(n) iv- x=0 k = “some constant” for i=1 to min(n, k) for j=1 to lg n x = x * lg n endfor endfor O(lgn) 3) 10 pts a) For each of the following recurrences, give an expression for the runtime T (n) if the recurrence can be solved with the Master Theorem. Otherwise, indicate that the Master Theorem does not apply....
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CS70%20Midterm_1_Fall%202008-%20Solution - CS570 Analysis...

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