Homework 4 Solutions

Homework 4 Solutions - CS180 Spring 2009 Homework 4...

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Unformatted text preview: CS180 Spring 2009 Homework 4 Solutions 4.9 (a) False. Counter-example: . The set { AB,BC,BD } is a valid minimum-bottleneck spanning tree but it is not a valid minimum spanning tree. (b) True. Suppose T is a minimum spanning tree of G , and T is a spanning tree with a lighter bottleneck edge. Add e max (the edge with maximal weight) in T to T . T now has a cycle, and by the cut property, e max does not belong to any minimum spanning tree, and thus cannot have been in T . 4.11 Label the edges arbitrarily from 1 to m , with the property that the last n- 1 belong to the desired T . Let δ be the minimum difference of two non-equal edge weights. For each edge i , e i ← e i- iδ n 3 . Now all edge weights are distinct, and the sorted order is the same as the originals. T is now the one with the most total weight reduction, and is now the unique MST of G , so Kruskal’s will return it. 4.29 First, remove from the list any vertices with degree d i = 0; these are isolated vertices in the graph. Otherwise, allgraph....
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This note was uploaded on 11/28/2009 for the course CS CS180 taught by Professor Mayer during the Spring '09 term at UCLA.

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