ShortestPath-handouts-2

ShortestPath-handouts-2 - Shortest Path 4/3/2006 2:42 PM...

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Shortest Path 4/3/2006 2:42 PM 1 Shortest Paths 1 Shortest Paths C B A E D F 0 3 2 8 58 4 8 7 1 2 5 2 3 9 Shortest Paths 2 Outline and Reading Weighted graphs (§7.1) ± Shortest path problem ± Shortest path properties Dijkstra’s algorithm (§7.1.1) ± Algorithm ± Edge relaxation The Bellman-Ford algorithm (§7.1.2) Shortest paths in dags (§7.1.3) All-pairs shortest paths (§7.2.1)
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Shortest Path 4/3/2006 2:42 PM 2 Shortest Paths 3 Weighted Graphs In a weighted graph, each edge has an associated numerical value, called the weight of the edge Edge weights may represent, distances, costs, etc. Example: ± In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports ORD PVD MIA DFW SFO LAX LGA HNL 849 8 0 2 1 3 87 17 4 9 1120 7 5 Shortest Paths 4 Shortest Path Problem Given a weighted graph and two vertices u and v , we want to find a path of minimum total weight between u and v. ± Length of a path is the sum of the weights of its edges. Example: ± Shortest path between Providence and Honolulu Applications ± Internet packet routing ± Flight reservations ± Driving directions ORD PVD MIA DFW SFO LAX LGA HNL
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Shortest Path 4/3/2006 2:42 PM 3 Shortest Paths 5 Shortest Path Properties Property 1: A subpath of a shortest path is itself a shortest path Property 2: There is a tree of shortest paths from a start vertex to all the other vertices Example: Tree of shortest paths from Providence ORD PVD MIA DFW SFO LAX LGA HNL 849 8 0 2 1 3 87 7 4 9 1120 5 Shortest Paths 6 Dijkstra’s Algorithm The distance of a vertex v from a vertex s is the length of a shortest path between s and v Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s Assumptions: ± the graph is connected ± the edges are undirected ± the edge weights are nonnegative We grow a “ cloud ” of vertices, beginning with s and eventually covering all the vertices We store with each vertex v a label d ( v ) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices At each step ± We add to the cloud the vertex u outside the cloud with the
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ShortestPath-handouts-2 - Shortest Path 4/3/2006 2:42 PM...

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