2.2 shortest paths

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2.2 Shortest Paths. Def : directed graph or digraph : of nodes, of arcs, functions associating to each , a tail and a head . A digraph is called simple if it has no loops or parallel (same directions) arcs. write for , forward arc, reverse arc in a path - PowerPoint PPT Presentation

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Linear Programming (Optimization)

Combinatorial Optimization 201212.2 Shortest PathsCombinatorial Optimization 20122Overview of resultsCombinatorial Optimization 20123Combinatorial Optimization 20124Combinatorial Optimization 20125Combinatorial Optimization 20126Combinatorial Optimization 20127Fords AlgorithmCombinatorial Optimization 20128Startvw= ravw= rbvw= advw= bavw= ady py py py py py pr0 00 00 00 00 00 0a -13 r3 r3 r2 b2 bb -1 -11 r1 r 1 r 1 rd -1 -1 -1 5 a5 a4 arbad1312(Example)Combinatorial Optimization 20129Combinatorial Optimization 201210Startvw= ravw= abvw= bdvw= davw= aby py py py py py pr0 00 00 00 00 00 0a -12 r2 r2 r1 d1 db -1 -13 a3 a3 a 2 ad -1 -1 -1 0 b0 b0 brdab-3211(Abnormal case)Combinatorial Optimization 201211Combinatorial Optimization 201212Combinatorial Optimization 201213Combinatorial Optimization 201214Combinatorial Optimization 201215Abnormal case (existence of negative cost dicircuit)If the network has a negative-cost dicircuit, the LP is unbounded, hence dual is infeasible.rs-5(M)1(M)2(M)Let flow of M on the dicircuit11Combinatorial Optimization 201216Validity of Fords AlgorithmCombinatorial Optimization 201217Combinatorial Optimization 201218Combinatorial Optimization 201219Combinatorial Optimization 201220Combinatorial Optimization 201221Feasible Potentials and Linear ProgrammingCombinatorial Optimization 201222Combinatorial Optimization 201223Combinatorial Optimization 201224Combinatorial Optimization 201225Combinatorial Optimization 201226The identification of the basis of the LP as a spanning tree (not necessarily arborescence) is used in the primal minimum cost flow algorithm for the minimum cost network flow problem. (Chapter 4)Combinatorial Optimization 201227Refinements of Fords AlgorithmCombinatorial Optimization 201228Combinatorial Optimization 201229The Ford-Bellman AlgorithmCombinatorial Optimization 201230Combinatorial Optimization 201231Combinatorial Optimization 201232Acyclic DigraphsCombinatorial Optimization 201233Nonnegative CostsCombinatorial Optimization 201234Combinatorial Optimization 201235Combinatorial Optimization 201236Unit Costs and Breadth-First SearchCombinatorial Optimization 201237Other Interpretation of Ford-Bellman Alg.Combinatorial Optimization 201238M Shortest Paths without Repeated Nodes