2Overview of the LectureA generic algorithm for solving shortest path problems znegative costs permittedzbut no negative cost cycle (at least for now)The use of reduced costs All pair shortest path problemINPUT G = (N, A) with costs cNode 1 is the source nodeThere is no negative cost cyclezWe will relax that assumption later
3Optimality ConditionsLemma. Let d*(j) be the shortest path length from node 1 to node j, for each j. Let d( ) be node labels with the following properties:d(j) ≤d(i) + cijfor i ∈N for j ≠1.(1)d(1) = 0.(2)Then d(j) ≤d*(j) for each j.Proof. Let P be any path from node 1 to node j, with length c(P), and suppose P has k arcs. Claim: d(j) ≤c(P). Note: if P is the shortest path from 1 to j, thend(j) ≤c(P) = d*(j), which is what we want to prove.
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