Singlesource shortest paths n Given a weighted graph

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Single-source shortest paths n Given a weighted graph G and a source vertex v

Single-source shortest paths n Given a weighted graph G and a source vertex v in G, determine the shortest paths from v to all other vertices in G n n 10/02/05 Path length: sum of all edges in path Useful in road map applications (e. g. , mapquest. com) for example Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s Algorithm n n n Solves the single-source shortest paths problem Involves keeping a

Dijkstra’s Algorithm n n n Solves the single-source shortest paths problem Involves keeping a table of current shortest path lengths from source vertex (initialize to infinity for all vertices except v, which has length 0) Repeatedly select the vertex u with shortest path length, and update other lengths by considering the path that passes through that vertex Stop when all vertices have been selected Could use a priority queue to facilitate selection of shortest lengths n n 10/02/05 Need to refine data structure so that the update of key-values in priority queue is allowed Time complexity: O( (n + m) log n ) or O( n 2 log n ), O( n 2 ) if computation of minimum is simplified Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm • 2704 ORD 849 740 621 1464 1235 DFW • 2342 BWI

Dijkstra’s algorithm • 2704 ORD 849 740 621 1464 1235 DFW • 2342 BWI 1090 946 1121 • 10/02/05 1258 1391 0 LAX • 184 337 • 144 JFK 802 SFO PVD 187 1846 • BOS 867 • MIA • Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm • 2704 ORD 849 740 621 1464 1235 DFW • 2342 BWI

Dijkstra’s algorithm • 2704 ORD 849 740 621 1464 1235 DFW • 2342 BWI 1090 946 1121 • 10/02/05 1258 1391 0 LAX 184 • 184 337 • 144 JFK 802 SFO PVD 187 1846 • BOS 867 621 • MIA 946 • Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm • 2704 ORD 849 740 621 1464 1235 DFW • 2342 BWI

Dijkstra’s algorithm • 2704 ORD 849 740 621 1464 1235 DFW • 2342 BWI 1090 946 1121 • 10/02/05 1258 1391 0 LAX 184 337 • 144 JFK 802 SFO PVD 187 1846 • BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm 371 2704 ORD 849 740 621 1464 1235 DFW 1575 2342 BWI

Dijkstra’s algorithm 371 2704 ORD 849 740 621 1464 1235 DFW 1575 2342 BWI 1090 946 1121 • 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 • BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm 371 2704 ORD 849 740 621 1464 1235 DFW 1575 2342 BWI

Dijkstra’s algorithm 371 2704 ORD 849 740 621 1464 1235 DFW 1575 2342 BWI 1090 946 1121 • 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 3075 • BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm 371 2704 ORD 849 740 621 1464 1235 DFW 1575 1423 2342

Dijkstra’s algorithm 371 2704 ORD 849 740 621 1464 1235 DFW 1575 1423 2342 BWI 1090 946 1121 • 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 3075 2467 BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342 BWI 1090 946 1121 3288 • 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 2467 BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342 BWI 1090 946 1121 3288 2658 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 2467 BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342 BWI 1090 946 1121 2658 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 2467 BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342

Dijkstra’s algorithm (cont) 371 2704 ORD 849 740 621 1464 1235 DFW 1423 2342 BWI 1090 946 1121 2658 10/02/05 1258 1391 0 LAX 184 337 328 144 JFK 802 SFO PVD 187 1846 2467 BOS 867 621 MIA 946 Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved Graphs Slide 10/02/05

SingleSource Shortest Paths SingleSource Shortest Path Problem GivenSingleSource Shortest Paths SingleSource Shortest Path Problem Given
A ForwardBackward SingleSource Shortest Paths Algorithm SingleSource ShortestA ForwardBackward SingleSource Shortest Paths Algorithm SingleSource Shortest
SingleSource Shortest Paths Problem Definition Shortest paths andSingleSource Shortest Paths Problem Definition Shortest paths and
Chapter 24 SingleSource Shortest Paths Shortest Paths HowChapter 24 SingleSource Shortest Paths Shortest Paths How
SINGLESOURCE SHORTEST PATHS ShortestPath Variants SingleSource SingleDestination 1SINGLESOURCE SHORTEST PATHS ShortestPath Variants SingleSource SingleDestination 1
Shortest Paths Weighted Graphs In a weighted graphShortest Paths Weighted Graphs In a weighted graph
Dijkstras Shortest Path Algorithm CutlerHead SingleSource Shortest PathsDijkstras Shortest Path Algorithm CutlerHead SingleSource Shortest Paths
Chapter 25 SingleSource Shortest Paths Problem Definition ShortestChapter 25 SingleSource Shortest Paths Problem Definition Shortest
4 4 SingleSource Shortest Paths Problem Definition Shortest4 4 SingleSource Shortest Paths Problem Definition Shortest
Page 620 SingleSource Shortest Paths Problem Definition ShortestPage 620 SingleSource Shortest Paths Problem Definition Shortest
Chapter 24 SingleSource Shortest Paths n Given AChapter 24 SingleSource Shortest Paths n Given A