GE330_lect14 - Lecture 14 Shortest Path Problems...

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Unformatted text preview: Lecture 14 Shortest Path Problems Dijkstra’s Algorithm, Max-Flow Problem March 16, 2009 Shortest-route problem Given a connected directed graph G ( N,A ) with a length (cost) c a associated with each a ∈ A , given a source node and a destination node, the shortest route problem is to find a route (path) from the source node to the destination node with the shortest length. (The length of a route is the sum of the lengths of the arcs on that route). For example, in the following graph, you want to find the shortest route from node 1 to node 8. 2 LP Formulation Given a network G ( N,A ), suppose we want to find the shortest route from s ∈ N to t ∈ N . We can artificially add one unit of supply to node s and one unit of demand to node t . Regard all other nodes as transhipment nodes. Let the cost on an arc be the length of the arc. We can then write down the LP formulation for the minimum cost flow problem on the network. Particularly, let x a be the flow on arc a . For a node i , let δ ( i ) + be the set of arcs entering i and δ ( i )- be the set of arcs leaving i . the model is: min X a ∈ A c a x a s.t. X a ∈ δ ( s )- x a- X a ∈ δ ( s ) + x a = 1 , X a ∈ δ ( t ) + x a- X a ∈ δ ( t )- x a = 1 , X a ∈ δ ( i ) + x a- X a ∈ δ ( i )- x a = 0 , ∀ i ∈ N- { s,t } x a ≥ , ∀ a ∈ A (1) 3 Shortest Path from one-to-all Given a network G = ( N, A ) we want to find the shortest path from one of the nodes to all other nodes in the network. Suppose N = { 1 , . . . , n } , and let c ij ≥ be a “length” of the arc ( i, j ) for ( i, j ) ∈ A . Without loss of generality, suppose we want to compute the shortest distance from 1 to all other nodes i in the network....
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This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE330_lect14 - Lecture 14 Shortest Path Problems...

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