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All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time

Authors Timothy M. Chan, Dimitrios Skrepetos

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Timothy M. Chan
Dimitrios Skrepetos

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Timothy M. Chan and Dimitrios Skrepetos. All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.24

Abstract

In this paper we study the all-pairs shortest paths problem in (unweighted) unit-disk graphs. The previous best solution for this problem required O(n^2 log n) time, by running the O(n log n)-time single-source shortest path algorithm of Cabello and Jejcic [Comput. Geom., 2015] from every source vertex,where n is the number of vertices. We not only manage to eliminate the logarithmic factor, but also obtain the first (slightly) subquadratic algorithm for the problem, running in O(n^2 sqrt{ frac{log log n}{log n} }) time. Our algorithm computes an implicit representation of all the shortest paths, and, in the same amount of time, can also compute the diameter of the graph.
Keywords
  • unit-disk graphs
  • all-pairs shortest paths
  • computational geometry

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