Approximate shortest paths in weighted graphs
Journal of Computer and System Sciences
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We show that for every 0 ≤ p ≤ 1 there is analgorithm with running time of O(n 2.575-p/(7.4 -2.3p)) that given a directed graphwith small positive integer weights, estimates the length of theshortest path between every pair of vertices u,vin the graph to within an additive error δp (u,v), whereδ(u,v) is the exact length of theshortest path between u and v. This algorithmruns faster than the fastest algorithm for computing exact shortestpaths for any 0 p ≤ 1.Previously the only way to "bit" the running time of the exactshortest path algorithms was by applying an algorithm of Zwick[FOCS '98] that approximates the shortest path distances within amultiplicative error of (1 + ε). Ouralgorithm thus gives a smooth qualitative and quantitativetransition between the fastest exact shortest pathsalgorithm, and the fastest approximation algorithm with alinear additive error. In fact, the main ingredient we need inorder to obtain the above result, which is also interesting in itsown right, is an algorithm for computing (1 + ε)multiplicative approximations for the shortest paths, whose runningtime is faster than the running time of Zwick's approximationalgorithm when ε