hw4s - Cs445 Homework#4 Graphs MST and Shortest Paths Due...

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Cs445 — Homework #4 Graphs, MST and Shortest Paths Due: 4/5/2005 during class meeting. All questions have the same weight. Assume that in all graphs discussed in this homework the number of edges is larger than the number of vertices. In all questions the notation | V | denotes the number of elements in the set V . Also assume that that for each graph G ( V, E ) discussed below, each edge ( u, v ) E is assigned a weight w ( u, v ) which is a real number. In all questions, the length of a path between vertices is the sum of weights of edges along the path, and for two vertices s, v , we denote by δ ( s, v ) the length of the path with minimum weight from s to v . 1. A graph G ( E, V ) is called a cycle graph if there is a path that visits all the edges and all the vertices exactly once, and starts and ends at the same vertex. Suggest an algorithm, as efficient as possible, for finding a Minimum Spanning Tree of a cycle graph. a b c d f e Figure 1: Example of a cycle graph Answer: Iterate through the set E , and find the edge with maximum weight. The graph obtained by removing this edge of highest weight, will be a MST. Only one edge needs to be removed. If you remove more than one edge, it no longer remains an MST as atleast one vertex will not be included. The runtime is linear since it takes linear time to find the maximum value. 1

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2. Let G ( V, E ) be an undirected graph. Suggest an algorithm that runs in O ( | E | ) time, and finds a spanning tree for G (not necessarily a minimum spanning tree). Answer: Solution 1 - ST = φ For each edge e E { if(adding e to ST doesnot introduce a cycle) add e to ST mark e } The test for a cycle can be done in constant time, by marking vertices that have been added to the ST. An edge will introduce a cycle, if both its vertices have already been marked.
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