HW1_solution_4 - ENGRI 115 Engineering Applications of OR...

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ENGRI 115 Engineering Applications of OR Fall 2007 Solution to Question 4 Homework 1 Question 4: Decide whether the following statements are TRUE or FALSE. If you think a statement is true, you need to justify it (in a few sentences). If you think a statement is false, give a specific example to show this. 1. Let G = ( V, E ) be an undirected graph, v V any node in G , and e E any “cheapest” (i.e., lowest cost) edge incident to v . Then, some minimum spanning tree contains e . 2. Let G = ( V, E ) be an undirected graph, v V any node in G , and T any minimum spanning tree in G . Then, T must contain some cheapest edge incident to v . Answer: 1. TRUE. The answer needs two arguments. We proved that Prim’s algorithm produces a minimum spanning tree (argument 1). Let’s focus on one of the cheapest edges incident to v , call it e (note: there can be several cheapest edges). Prim’s algorithm can be started at node v , and it can choose e as first edge (since ties can be broken arbitrarily) (argument 2). Thus, the MST produced by Prim
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This note was uploaded on 06/01/2008 for the course ENGRI 1101 taught by Professor Trotter during the Fall '05 term at Cornell University (Engineering School).

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