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- W2950680090 abstract "We introduce a new problem that combines the well known All Pairs Shortest Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to compute the shortest paths for all pairs of vertices for all possible flow amounts. We call this new problem the All Pairs Shortest Paths for All Flows (APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with unit edge costs and real edge capacities in $tilde{O}(sqrt{t}n^{(omega+9)/4}) = tilde{O}(sqrt{t}n^{2.843})$ time, where $n$ is the number of vertices, $t$ is the number of distinct edge capacities (flow amounts) and $O(n^{omega}) < O(n^{2.373})$ is the time taken to multiply two $n$-by-$n$ matrices over a ring. Secondly we extend the problem to graphs with positive integer edge costs and present an algorithm with $tilde{O}(sqrt{t}c^{(omega+5)/4}n^{(omega+9)/4}) = tilde{O}(sqrt{t}c^{1.843}n^{2.843})$ worst case time complexity, where $c$ is the upper bound on edge costs." @default.
- W2950680090 created "2019-06-27" @default.
- W2950680090 creator A5007980325 @default.
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- W2950680090 date "2013-09-22" @default.
- W2950680090 modified "2023-09-23" @default.
- W2950680090 title "Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems" @default.
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- W2950680090 doi "https://doi.org/10.48550/arxiv.1309.5687" @default.
- W2950680090 hasPublicationYear "2013" @default.
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