Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths

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Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths (1993)

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by David R. Karger , Daphne Koller , Steven J. Phillips

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@MISC{Karger93findingthe,
    author = {David R. Karger and Daphne Koller and Steven J. Phillips},
    title = {Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths},
    year = {1993}
}

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Abstract

We investigate the all-pairs shortest paths problem in weighted graphs. We present an algorithm---the Hidden Paths Algorithm---that finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any path-comparison based algorithm for the all-pairs shortest paths problem. Path-comparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.

Keyphrases

all-pairs shortest path time bound hidden path all-pairs shortest path problem hidden path algorithm path algorithm natural class shortest path weighted graph algorithm form path problem edge weight many probability distribution general setting generalized form high probability practical substitute running time

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