hw8solns

# hw8solns - Discrete Mathematics Spring 2004 Homework 8...

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Unformatted text preview: Discrete Mathematics, Spring 2004 Homework 8 Sample Solutions 6.4 #5. Find the length of a shortest path and a shortest path between the vertices h and d in the following graph: 2 7 6 5 4 4 6 3 5 6 2 a c d b z f e 5 2 7 4 7 4 3 h i j g 5 2 4 3 Solution . The shortest path is h , f , c , d , with total length 10. To see this, we use Dijkstra’s algorithm, modified as shown in Example 6.4.4 in the text. Let V ( j ) denote the vertex chosen in the beginning of the j th iteration of the while loop, let L ( j ) denote the weight of the least weight path from h to V ( j ), and let P ( j ) denote the vertex from which V ( j ) was labeled. The table below indicates the order of selection; the last column of the table indicates the vertices which have not yet been made permanent at the end of the j th iteration (but have weight < ∞ ). j V ( j ) L ( j ) P ( j ) Temporary ( V,L,P ) 1 h ∅ ( a, 4 ,h ) , ( e, 7 ,h ) , ( f, 5 ,h ) , ( i, 2 ,h ) 2 i 2 h ( a, 4 ,h ) , ( e, 7 ,h ) , ( f, 5 ,h ) , ( j, 8 ,i ) 3 a 4 h ( e, 7 ,h ) , ( f, 5 ,h ) , ( j, 8 ,i ) , ( b, 7 ,a ) 4 f 5 h ( e, 7 ,h ) , ( j, 8 ,i ) , ( b, 7 ,a ) , ( c, 7 ,f ) , ( g, 9 ,f ) 5 b 7 a ( e, 7 ,h ) , ( j, 8 ,i ) , ( c, 7 ,f ) , ( g, 9 ,f ) 6 c 7 f ( e, 7 ,h ) , ( j, 8 ,i ) , ( g, 9 ,f ) , ( d, 10 ,c ) 7 e 7 h ( j, 8 ,i ) , ( g, 9 ,f ) , ( d, 10 ,c ) 8 j 8 i ( g, 9 ,f ) , ( d, 10 ,c ) , ( z, 13 ,j ) 9 g 9 f ( d, 10 ,c ) , ( z, 13 ,j ) 10 d 10 c done 1 6.4 #8. Write an algorithm that finds the lengths of the shortest paths between all vertex pairs in a simple, connected, weighted graph having n vertices in time O ( n 3 ). Solution . Probably the easiest approach is to take Dijkstra’s algorithm as shown in the text, and then make a few minor modifications. The input to our algorithm will now simply be the n × n matrix w of edge weights of a simple connected graph G whose vertices are labeled 1 to n . (For convenience, we assume that w ( i,j ) = ∞ if i and j are not adjacent.) The output will be the n × n matrix L...
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hw8solns - Discrete Mathematics Spring 2004 Homework 8...

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