All-Pairs Bottleneck Paths in Vertex Weighted Graphs

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All-Pairs Bottleneck Paths in Vertex Weighted Graphs (2007)

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by Asaf Shapira , Raphael Yuster , Uri Zwick

Venue:	In Proc. of SODA, 978–985
Citations:	9 - 1 self

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@INPROCEEDINGS{Shapira07all-pairsbottleneck,
    author = {Asaf Shapira and Raphael Yuster and Uri Zwick},
    title = {All-Pairs Bottleneck Paths in Vertex Weighted Graphs},
    booktitle = {In Proc. of SODA, 978–985},
    year = {2007},
    pages = {978--985}
}

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Abstract

Let G = (V, E, w) be a directed graph, where w: V → R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n 2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights. The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on

Keyphrases

all-pairs bottleneck path apbp problem vertex weighted graph bottleneck weight ordered pair directed graph all-pairs shortest path asymptotic loss maximum weight arbitrary weight function williams stoc maximum bottleneck weight fast matrix multiplication numerous application first sub-cubic algorithm arbitrary large vertex weight integer edge time algorithm sub-cubic algorithm main result several interesting problem vertex weight

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