Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems

  • Tong-Wook Shinn
  • Tadao Takaoka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


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 ω ) < 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.


Short Path Acceleration Phase Path Cost Successor Node Edge Cost 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Tong-Wook Shinn
    • 1
  • Tadao Takaoka
    • 1
  1. 1.Department of Computer Science and Software EngineeringUniversity of CanterburyChristchurchNew Zealand

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