Homework2 - ENGRI 1101 Engineering Applications of OR Fall...

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ENGRI 1101 Engineering Applications of OR Fall ’09 Homework 2 The Shortest Path Problem Due date: 4pm on September 16, 2009 in Rhodes ENGRI 1101 dropbox. Reading assignment: Handout 3 on the shortest path problem. Be sure to write which lab you attend and your NETID at the top of your homework. 1. (70) Suppose that you are given the graph G = ( N, A ) with N = { 1 , 2 , 3 , 4 , 5 } and A = { (1 , 2) , (1 , 3) , (2 , 1) , (2 , 3) , (2 , 5) , (3 , 1) , (3 , 4) , (4 , 2) , (5 , 3) , (5 , 4) } and a special source node s = 6 . The lengths of these arcs are given in the table below: ( i, j ) ( i, j ) (1,2) 2 (1,3) 6 (2,1) 3 (2,3) 4 (2,5) 3 (3,1) 2 (3,4) 1 (4,2) 6 (5,3) 1 (5,4) 5 Be sure to explain ALL of your work. (a) (5) Draw a picture of the directed graph G . (b) (5) Is (1 , 2) , (2 , 3) , (3 , 5) a path from node 1 to node 5 in this graph? Why or why not? (c) (5) Is (4 , 3) N ? is (4 , 3) A ? (d) (20) Compute the length of the shortest path from the source node 5 to each other node. (e) (5) Compute the shortest path (the path, not its length) from node 5 to node 2. (f) (10) Compute the shortest path tree for this input.
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