More algorithms for all-pairs shortest paths in weighted graphs

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More algorithms for all-pairs shortest paths in weighted graphs (2007)

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by Timothy M. Chan

Venue:	In Proceedings of 39th Annual ACM Symposium on Theory of Computing
Citations:	74 - 3 self

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@INPROCEEDINGS{Chan07morealgorithms,
    author = {Timothy M. Chan},
    title = {More algorithms for all-pairs shortest paths in weighted graphs},
    booktitle = {In Proceedings of 39th Annual ACM Symposium on Theory of Computing},
    year = {2007},
    pages = {590--598}
}

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Abstract

In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general real-weighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs. 1

Keyphrases

all-pairs shortest path weighted graph small-integer-weighted graph second part general real-weighted dense graph first part subcubic apsp algorithm time bound path problem constant dimension new algorithm log log subcubic result euclidean space large class real-vertex-weighted graph running time improved result fast matrix multiplication all-pairs lightest

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