A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

  • Andrzej Lingas
  • Dzmitry Sledneu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7147)

Abstract

We consider the problem of computing all-pairs shortest paths in a directed graph with non-negative real weights assigned to vertices.

For an n×n 0 − 1 matrix C, let K C be the complete weighted graph on the rows of C where the weight of an edge between two rows is equal to their Hamming distance. Let MWT(C) be the weight of a minimum weight spanning tree of K C .

We show that the all-pairs shortest path problem for a directed graph G on n vertices with non-negative real weights and adjacency matrix A G can be solved by a combinatorial randomized algorithm in time
$$\widetilde{O}(n^{2}\sqrt{n + \min\{MWT(A_G), MWT(A_G^t)\}})$$
As a corollary, we conclude that the transitive closure of a directed graph G can be computed by a combinatorial randomized algorithm in the aforementioned time.

We also conclude that the all-pairs shortest path problem for vertex-weighted uniform disk graphs induced by point sets of bounded density within a unit square can be solved in time \(\widetilde{O}(n^{2.75})\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andrzej Lingas
    • 1
  • Dzmitry Sledneu
    • 2
  1. 1.Department of Computer ScienceLund UniversityLundSweden
  2. 2.The Centre for Mathematical SciencesLund UniversityLundSweden

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