3510: Name:
CS Design and Analysis of Algorithms Summer 2008 Quiz II
Instructions: Please write your name in the space provided at the upper left corner of this page. This quiz has 5 questions on the following 5 pages. For each question, write your answer in the space provided for it. You may write on the back of a page if the space provided is not sufficient. GOOD LUCK!
Question 1 2 3 4 5 Total
Score
0
Problem 1
If the input graph is given by its adjacency matrix, what would the running time of each of the following algorithms be? Briefly justify your answer. 1. Depth-First Search Answer: O(n2 )
2. Kruskal's Algorithm Answer: O(n2 + m log n)
3. Bellman-Ford Algorithm Answer: O(n2 + nm)
4. Dijkstra's Algorithm Answer: O(n2 )
5. Floyd-Warshall Algorithm Answer: O(n3 )
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Problem 2
Indicate whether the following statement is true or false: Let G be a connected, weighted, undirected graph and let T be a minimum spanning tree of G. If the weight of one edge in T is decreased and the weights of all other edges in G are unchanged, then T remains a minimum spanning tree of the resulting new graph. Justify your answer. Answer: The statement is true. Let w and w be the old and new weight functions respectively. Let e be the edge in T whose weight is decreased. Thus w (e) = w(e) - k for some k > 0, and w(e ) = w (e ) for each edge e = e. Let T be any other spanning tree of G. Since T is a MST of G with respect to the old weight function w, w(T ) w(T ). If e T , then w (T ) < w(T ) w(T ) = w (T ). If e T , then w (T ) = w(T ) - k w(T ) - k = w (T ). Thus T remains a MST after the weight change.
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Problem 3
Suppose that we are given a weighted undirected graph G = (V, E), a source s V , a destination t V , and a subset W V of vertices, and want to find a shortest path from s to t that must go through at least one vertex in W . Design an efficient algorithm that solves the problem by only two applications of a given algorithm for single-source shortest paths. Show that your algorithm is correct. Solution: Since the graph is undirected, for every pair of vertices u and v, a shortest path from u to is v also a shortest path from v to u due to symmetry. It can be seen that the weight of a shortest path from s to v involving at least one vertex in w W , equals
wW
min {(s.w) + (t, w)} .
Let SP be any given algorithm for single-source shortest paths. Then the following algorithm solves the problem with two applications of SP: 1. Run SP(G, s) (using s as the source). 2. Run SP(G, t) (using t as the source). 3. Find a vertex w0 W with the smallest value of (s.w0 ) + (t, w0 ). 4. Return the concatenation of the shortest path between s and w0 , and that between t and w0 .
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Problem 4
Let G = (V, E) be a weighted directed graph whose all-pairs shortest paths have already been computed. Suppose that a new edge is added to G. Show how to update the all-pairs shortest paths in O(n2 ) time, where n = |V |. Solution: Suppose a new edge (x, y) is added. For each pair of vertices u, v, du,v min {du,v , du,x + w(x, y) + dy,v } .
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Problem 5
In class we learned an algorithm for computing a longest common subsequence of two strings. Call this algorithm 2-LCS. Now suppose that we want to compute a longest common subsequence of three strings. Consider a greedy algorithm 3-LCS(x, y, z) that simply outputs 2-LCS(2-LCS(x, y), z). Does 3-LCS correctly compute a longest common ...