Lecture 18 Solving Shortest Path Problem: Dijkstra’s Algorithm
Lecture 18Solving Shortest Path Problem: Dijkstra s AlgorithmOctober 23, 2009Lecture 18Outline Focus on Dijkstra s Algorithm Importance: Where it has been used? Algorithm s general description Algorithm steps in detail ExampleOperations Research Methods1Lecture 18One-To-All Shortest Path ProblemWe are given a weighted network(V, E, C)with node setV, edge setE,and the weight setCspecifying weightscijfor the edges(i, j) E. We arealso given a starting nodes V. Theone-to-all Shortest pathproblem isthe problem of determining the Shortest path from nodesto all the othernodes in the weights on the links are also referred Research Methods2Lecture 18Algorithms Solving the Problem Dijkstra s Algorithm Solves only the problems with nonnegative costs, ,cij 0for all(i, j) E Bellman-Ford Algorithm Applicable to problems with arbitrary costs Floyd-Warshall Algorithm Applicable to problems with arbitrary costs Solves a more general all-to-all Shortest path problemFloyd-Warshall and Bellman-Ford Algorithm solve the problems on graphsthat do not have a cycle with negative Research Methods3Lecture 18Importance of Dijkstra s algorithmMany more problems than you might at first think can be cast as shortestpath problems, making Dijkstra s Algorithm a powerful and general example: Dijkstra s Algorithm is applied to automatically find directions betweenphysical locations
Lecture 18 Algorithms Solving the Problem • Dijkstra’s algorithm • Solves only the problems with nonnegative costs, i.e., c ij ≥ 0 for all (i,j) ∈ E • Bellman-Ford algorithm • Applicable to problems with arbitrary costs • Floyd-Warshall algorithm • Applicable to problems with arbitrary costs • Solves a more general all-to-all shortest path problem ...
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