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Handout3

# Handout3 - ENGRI 1101 Engineering Applications of OR Fall...

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ENGRI 1101 Engineering Applications of OR Fall 2008 Handout 3 The shortest path problem Consider the following problem. You are given a map of the city in which you live, and you wish to figure out the fastest route to travel from your home to your office. In your city, some of the streets are two-way, and some are one-way. Furthermore, traveling down a street in one direction might not take the same time as in the other direction (e.g, if there is some construction taking place on your side of the street). First of all, we would like to give a mathematical model of this problem. To do this, it will be useful to introduce the notion of a directed graph . A directed graph consists of a set of nodes, and a set of arcs. For example, the picture below shows a graph in which 1, 2, 3, 4, 5, and 6 are the nodes of the graph. That is, in drawing a graph we represent a node by a circle with its name indicated inside. An arc is an ordered pair of nodes, such as (1 , 2) . The arc (1 , 2) is represented below as the arrow that points from node 1 to node 2. For nodes 2 and 3, there is an arc from 2 to 3 and an arc from 3 to 2. Thus, if we consider the graph below, then the set of nodes is { 1 , 2 , 3 , 4 , 5 , 6 } and the set of arcs is { (1 , 2) , (1 , 3) , (2 , 3) , (2 , 4) , (3 , 2) , (3 , 5) , (4 , 3) , (4 , 6) , (5 , 2) , (5 , 6) } . If we let N be the name for the set of nodes, that is, N = { 1 , 2 , 3 , 4 , 5 , 6 } , and if we let A be the name for the set of arcs then A = { (1 , 2) , (1 , 3) , (2 , 3) , (2 , 4) , (3 , 2) , (3 , 5) , (4 , 3) , (4 , 6) , (5 , 2) , (5 , 6) } . When we specify the elements that are contained in a set, then it does not matter in which order we list them. So for example, we could equally well have described N as { 1 , 3 , 4 , 6 , 5 , 2 } ; that is the same set. A graph consists of a set of nodes and a set of arcs; hence, if we call the graph G , then we often write that G = ( N,A ) to mean that N is its set of nodes, and A is its set of arcs. 3 5 6 1 2 4 Figure 1: A graph with 6 nodes and 10 arcs A path in a graph is a sequence of arcs that, from a visual perspective, you could follow with your pencil without lifting the pencil up. For example, (2 , 3) , (3 , 5) , (5 , 6) is a path from node 2 to node 6 in the graph given in Figure 1. There are two important things to notice. First, a path is a sequence of arcs, not a set of arcs: the order in which we list the arcs does matter. Second, we are following each arc in its given direction. For example, (3 , 2) , (2 , 1) is not a path from node 3 to node 1, since there is no arc (2 , 1) in the graph in Figure 1; only (1 , 2) is an arc in this graph. In general, we can write a path as follows: let i 1 , i 2 , . . . , i k 1

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denote nodes in the graph (not necessarily the nodes 1 , 2 ,...,k ); then ( i 1 ,i 2 ) , ( i 2 ,i 3 ) , ( i 3 ,i 4 ) ,..., ( i k - 1 ,i k ) is a path in the graph from node i 1 to node i k provided that each of ( i 1 ,i 2 ) , ( i 2 ,i 3 ) through ( i k - 1 ,i k ) is an arc in the graph. This path has k - 1 arcs in it. We will often be interested in directed graphs for which each arc has an associated length. We will denote the length of each arc ( i,j ) in A by ( i,j ) . In the graph below, we have added lengths by writing each arc’s length right next to it. For example, the length of arc (3 , 2) is 5, or equivalently, (3 , 2) = 5 . The length of a path is the sum of the lengths of the arcs in it.
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Handout3 - ENGRI 1101 Engineering Applications of OR Fall...

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