lec14 - Networks Lecture 1 Amedeo R Odoni Thanks to Prof R...

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Networks: Lecture 1 Amedeo R. Odoni November 15, 2006 * Thanks to Prof. R. C. Larson for some of the slides General Comments From continuous to a more “discretized” travel environment Enormous literature and variety of problems Transportation and logistics, urban services just two of the major areas of applications Level of detail of model depends on problem Numerous interpretations of “nodes” (“points”, “vertices”) and “arcs” (“links”, “edges”) Will concentrate on routing and location problems Will assume that efficient shortest path algorithms are available

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Outline and References Introduction Minimum Spanning Tree (MST) Chinese Postman Problem (CPP) Skim Sections 6.1 and 6.2, read Sections 6.3- 6.4.4 in Larson and Odoni Far more detailed coverage in (among others) Ahuja, R., T. L. Magnanti and J. B. Orlin, Network Flows, Prentice-Hall, 1993. Network with Terminology A B C D E Nodes B and D Directed Arc Undirected Arc
Examples of Nodes & Arcs Nodes/ Vertices/ Points Street intersections Towns Cities Electrical junctions Project milestones Arcs/ Edges/ Links Street segments Country roads Airplane travel time Circuit components Project tasks Network Terminology N = sets of nodes A = set of arcs G(N,A) Incident arc Adjacent nodes Adjacent arcs Path Degree of a node In-degree Out-degree Cycle or circuit Connected nodes Connected undirected graph Strongly connected directed graph Subgraph

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Network Terminology - con't. Tree of an Spanning tree of undirected network G(N,A) is a tree is a connected containing all n subgraph having no nodes of N cycles Length of a path S A tree having t L ( S ) = l ( i , j ) nodes contains (t-1) ( i , j ) S edges d(x,y), d(i,j) Shortest Path Problem Find the shortest path (more generally, least cost path) between two nodes, starting at Node O and ending at Node D. Dijkstra’s node labeling algorithm (essentially dynamic programming); one-to-all paths; all edge lengths are non-negative; O(n 2 ). Floyd’s algorithm; negative edge lengths OK (discovers negative cycles); all-to-all paths; non-obvious; O(n 3 ). Numerous variations and extensions: all-to-one; critical
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lec14 - Networks Lecture 1 Amedeo R Odoni Thanks to Prof R...

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