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lecture20

# lecture20 - Shortest Paths The Algorithms Shortest Paths...

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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 Paths—Definitions 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 0 , 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 δ ( u, v ) . We use the convention that δ ( 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

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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 shortest
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lecture20 - Shortest Paths The Algorithms Shortest Paths...

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