lecture20 - Shortest Paths The Algorithms IE170: Algorithms...

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Unformatted text preview: Shortest Paths The Algorithms IE170: Algorithms in Systems Engineering: Lecture 20 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University March 19, 2007 Jeff Linderoth IE170:Lecture 20 Shortest Paths The Algorithms Taking Stock Last Time Minimum Spanning Trees Strongly Connected Components This Time Shortest Paths Jeff Linderoth IE170:Lecture 20 Shortest Paths The Algorithms Shortest Path Properties Shortest PathsDefinitions For the next few lectures, we will have a directed graph G = ( V,E ) , and a weight function w : E R | E | . The weight of a path P = { v ,v 1 ,. .. v k } is simply the weight of the edges taken on the sequence of nodes: w ( P ) = k i =1 w v i- 1 ,v i . We are interested in finding the shortest-path weights from u to v , which we will denote ( u, v ) . We use the convention that ( u, v ) = if there is no path from u to v in G Jeff Linderoth IE170:Lecture 20 Shortest Paths The Algorithms Shortest Path Properties Example The example (hopefully) makes it clear that shortest paths are organized as a tree Many algorithms work like a generalization of BFS to weighted graphs. Jeff Linderoth IE170:Lecture 20 Shortest Paths The Algorithms Shortest Path Properties Shortest Path Variants Single-Source : Find the shortest path from s V to every vertex v V Single-Destination : Find the shortest path from every vertex v V to a given destination vertex t V Single-Pair : Find the shortest path from given s V to given t V . There is now way known that is better (in the worst case) that solving the single-source version. All-Pairs : Find the shortest path from every u V to every vertex v V Jeff Linderoth IE170:Lecture 20 Shortest Paths The Algorithms Shortest Path Properties Negative Weight Edges In Minimum Spanning Tree, negative weight edges posed no significant challenge to the algorithms. However, for shortestsignificant challenge to the algorithms....
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This note was uploaded on 08/06/2008 for the course IE 170 taught by Professor Ralphs during the Spring '07 term at Lehigh University .

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lecture20 - Shortest Paths The Algorithms IE170: Algorithms...

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