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Unformatted text preview: Lecture 41 SSSP with negative edge weights 20.2.4 SingleSource AllDestinations Shortest Paths With Negative Costs • Directed weighted graph. • Edges may have negative cost. • No cycle whose cost is < 0 . • Find a shortest path from a given source vertex s to each of the n vertices of the digraph. SingleSource AllDestinations Shortest Paths With Negative Costs • Dijkstra’s O(n 2 ) singlesource greedy algorithm doesn’t work when there are negativecost edges. • Floyd’s Θ (n 3 ) allpairs dynamicprogramming algorithm does work in this case. BellmanFord Algorithm • Singlesource alldestinations shortest paths in digraphs with negativecost edges. • Uses dynamic programming. • Runs in O(n 3 ) time when adjacency matrices are used. • Runs in O(ne) time when adjacency lists are used. Decision Sequence • To construct a shortest path from the source to vertex v , decide on – the max number of edges on the path and on – the vertex that comes just before v . • Since the digraph has no cycle whose length is < 0 , we may limit ourselves to the discovery of cycle free (acyclic) shortest paths....
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 Fall '08
 Davis
 Graph Theory, Shortest path problem, shortest path, Digraphs

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