Abstract
In this paper, we address the problem of finding the [Formula: see text] best paths connecting a given pair of nodes in a directed graph with arbitrary lengths. We introduce an algorithm to determine the [Formula: see text] best paths in order of length when repeat nodes in the paths are allowed. We obtain an O[Formula: see text] time and O[Formula: see text] space algorithm to implicitly determine the [Formula: see text] shortest paths in a directed graph with [Formula: see text] nodes and [Formula: see text] arcs. Empirical results where the algorithm was used to compute one hundred million paths in USA road networks are reported. A non-trivial modification of the previous algorithm is performed obtaining an O[Formula: see text] time method to compute paths without repeat nodes and to answer the next question: how many paths [Formula: see text] in practice are needed to identify [Formula: see text] simple paths using the previous algorithm? We find that the response is usually O[Formula: see text] based on an experiment computing one million paths in USA road networks. The determination of a theoretical tight bound on [Formula: see text] remains as an open problem.
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