6417c09 - Shortest Paths CONTENTS Introduction to Shortest...

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1 Shortest Paths CONTENTS Introduction to Shortest Paths (Section 4.1) Applications of Shortest Paths (Section 4.2) Network Simplex Algorithm (Section 11.7) Optimality Conditions (Section 5.2) Generic Label-Correcting Algorithm (Section 5.3) Specific Implementations (Section 5.4) Detecting Negative Cycles (Section 5.5) Shortest Paths in Acyclic Networks (Section 4.4) Dijkstra’s Algorithm (Section 4.5) Heap Implementations (Section 4.7) Dial’s Implementation (Section 4.6) Radix-Heap Implementation (Section 4.8)

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2 Problem Definition Shortest Path Problem : Identify a shortest path from the source node s to the sink node t in a directed network with arc costs (or lengths) given by c ij ’s. PATH LENGTH 1-2-4-6 60 1-2-4-5-6 90 1-2-5-6 45 1-2-3-5-6 100 1-3-5-6 80 Shortest Path: 1-2-5-6 1 2 4 3 5 6 10 25 35 20 15 35 40 30 20 s t
3 An Alternative Problem Identify a shortest path from the source node s to every other node in a directed network. In the process of determining shortest path from node s to node t, we also determine shortest path from node s to every other node in the network, which is specified by a directed out-tree. We shall henceforth consider this more general problem. 1 2 4 3 5 6 10 25 35 20 15 35 40 30 20 s

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4 A Linear Programming Problem The shortest path problem can be conceived of as a minimum cost flow problem: Send unit flow from the source node s to every other node in the network. Minimize subject to x ij 0 for every arc (i, j) A c x ij (i,j) A ij x x (n 1) for i s, 1 for all i N {s}, ij ji {j:(j,i) A} {j:(i,j) A} - = - = - = - R S T
5 Assumptions All arc costs (or lengths) c ij 's are integer and C is the largest magnitudes of arc costs. Rational arc costs can be converted to integer arc costs. Irrational arc costs (such as, ) cannot be handled. The network contains a directed path from node s to every node in the network. We can add artificial arcs of large cost to satisfy this assumption. The network does not contain a negative cycle. In the presence of negative cycles, the optimal solution of the shortest path problem is unbounded. The network is directed. To satisfy this assumption, replace each undirected arc (i, j) with cost c ij by two directed arc (i, j) and (j, i) with cost c ij . 2

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6 Applications of Shortest Paths Find a path of minimum length in a network. Find a path taking minimum time in a network. In a network G with arc reliabilities given by r ij ’s, find a path P of maximum reliability (given by Π (i,j) P r ij ). As a subroutine in a multitude of problems: Minimum cost flow problem Multi-commodity flow problems Network design problems
7 Determining Optimal Rental Policy Beverly owns a vacation rental which is available for rent for three months, say May 1 to July 31.

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6417c09 - Shortest Paths CONTENTS Introduction to Shortest...

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