Shortest Path Centrality and the All-pairs Shortest Paths Problem via Sample Complexity
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In this paper we are interested in the all-pairs shortest paths problem (APSP) for an input graph G assumed to be connected undirected with non negative real edge weights. In the exact deterministic case, it is an open question whether this problem admits a O(n^3-c) time algorithm, for any constant c>0, even in the case where edge weights are natural numbers. There is a variety of approximation algorithms for the problem and the time complexity of the fastest one depends on the graph being sparse or dense and also on the corresponding approximation guarantee. In this paper we deal with a version of the APSP that fits neither in the exact nor the approximate case. We give a O(n^2 log n) randomized algorithm for APSP such that for every pair of vertices the algorithm either computes the exact shortest path or does not compute any shortest path, depending on a certain measure of “importance” (or centrality) of the shortest path in question.
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