Figure 1515 A simple main

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Unweighted Shortest-Path Problem

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1 / / Process a request; return false if end of file 2 boo1 processRequest( istream & in, Graph & g ) 3 i 4 string startName, destName; 5 cout << "Enter start node: " ; 6 7 if ( ! ( in >> startName ) ) 8 return false; 9 cout << "Enter destination node: " ; 10 if ( ! ( in >> destName ) ) 11 r e ~ u r nfalse; 12 13 try 14 i 15 gnegative( startName ) ; 16 gprintPath( destName ) ; 17 } 18 catch( const GraphException h e ) 19 I 20 cerr < < etostring( ) < < endl; 21 } 22 return true; 23 1

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Figure 1516 For testing purposes, processRequest calls one of the shortestpath algorithms

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Once the graph has been read we repeatedly call processReques t , shown in Figure 1516 This version (which is simplified slightly from the online code) prompts for a starting and ending vertex and then calls one of the shortest-path algorithms This algorithm throws a GraphException if for instance, it is asked for a path between vertices that are not in the graph Thus processRequest catches any GraphException that might be generated and prints an appropriate error message

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The '"weighredpath length measures the of edges on a path

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Recall that the unweighted path length measures the number of edges In this section we consider the problem of finding the shortest unweighted path length between specified vertices

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UNWEIGHTED SINGLE-SOURCE, SHORTEST-PATH PROBLEM FIXDTHE SHORTEST PATH (,WEASL'RED BY NLMBER OF EDGES) FROM A DESIG!!ATED VERTEX S TO EVERY VERTEX

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The unweighted shortest-path problem is a special case of the weighted shortest-path problem (in which all weights are 1) Hence it should have a more efficient solution than the weighted shortest-path problem That turns out to be true, although the algorithms for all the path problems are similar

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AII variations of the shortest-path problem have similar solutions

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TO solve the unweighted shortest-path problem, we use the graph previously shown in Figure 151, with V 2 as the starting vertex S For now, we are concerned with finding the length of all shortest paths Later we maintain the corresponding paths W can see immediately that the shortest path from S to V 2is a path of ; length 0 This information yields the graph shown in Figure 1517 Now we can start looking for all vertices that are distance 1 from S We can find them by looking at the vertices adjacent to S If we do so, we see that V o and V 5 are one edge away from S, as shown in Figure 1518

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Figure 1517 The graph, after the starting vertex has been marked as reachable in zero edges

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Figure 1518 The graph, after all the vertices whose path length from the starting vertex is 1 have been found

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Next, we find each vertex whose shortest path from S is exactly 2 We do so by finding all the vertices adjacent to V oor V , (the vertices at distance 1) whose shortest paths are not already known This search tells us that the shortest path to V1 and V 3 is 2 Figure 1519 shows our progress so far Finally, by examining the vertices adjacent to the recently evaluated V , and V 3 ,we find that V , and V , have a shortest path of 3 edges All vertices have now been calculated Figure 1520 shows the final result of the algorithm This strategy for searching a graph is called breadth-first search, which operates by processing vertices in layers: Those closest to the start are evaluated first, and those most distant are evaluated last Figure 1521 illustrates a fundamental principle: If a path to vertex v has = + cost D,, and w is adjacent to v, then there exists a path to w of cost D, D, 1 All the shortest-path algorithms work by starting with D, = oo and reducing its value when an appropriate v is scanned To do this task efficiently, we must scan vertices v systematically When a given v is scanned, we update the vertices w adjacent to v by scanning through v's adjacency list From the preceding discussion, we conclude that an algorithm for solving the unweighted shortest-path problem is as follows Let D, be the length of the shortest path from S to i We know that D s= 0 and initially = that D i oo for all i # S We maintain a roving eyeball that hops from vertex

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