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# spneg - Shortest paths in graphs with negative edge weights...

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Shortest paths in graphs with negative edge weights Let G = ( V, E ) be a directed graph with edge weights c uv for each edge ( u, v ) E . Let | V | = n and | E | = m . Cost c ( p ) of a path p = ( v 0 , v 1 , . . . , v k ) is k i =1 c v i - 1 ,v i . Shortest path from s to t is a path with minimum cost among all paths from s to t . Problem: find a shortest path from s to all vertices. Need the following fact: – FACT 1: If G has no negative cycles then there is a shortest path from s to t with at most n - 1 edges. 1

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Bellman-Ford Algorithm Works in the presence of negative weight edges but no negative cycles. Let opt ( i, v ) be the minimum cost of a path from v to t using at most i edges. To compute opt ( n - 1 , s ). Let P be an optimal path with cost opt ( i, v ). There are two mutually exclusive cases: CASE 1: P uses at most i - 1 edges: * opt ( i, v ) = opt ( i - 1 , v ). CASE 2: P uses i edges: * Let the first edge of P be ( v, w ). * opt ( i, v ) = c vw + opt ( i - 1 , w ).
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