WADS 2017: Algorithms and Data Structures pp 253-264

# All-Pairs Shortest Paths in Geometric Intersection Graphs

• Timothy M. Chan
• Dimitrios Skrepetos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

## Abstract

We address the All-Pairs Shortest Paths (APSP) problem for a number of unweighted, undirected geometric intersection graphs. We present a general reduction of the problem to static, offline intersection searching (specifically detection). As a consequence, we can solve APSP for intersection graphs of n arbitrary disks in $$O\left( n^2\log n\right)$$ time, axis-aligned line segments in $$O\left( n^2\log {\log n}\right)$$ time, arbitrary line segments in $$O\left( n^{7/3}\log ^{1/3} n\right)$$ time, d-dimensional axis-aligned boxes in $$O\left( n^2\log ^{d-1.5} n\right)$$ time for $$d\ge 2$$, and d-dimensional axis-aligned unit hypercubes in $$O\left( n^2\log {\log n}\right)$$ time for $$d=3$$ and $$O\left( n^2\log ^{d-3} n\right)$$ time for $$d\ge 4$$.

In addition, we show how to solve the Single-Source Shortest Paths (SSSP) problem in unweighted intersection graphs of axis-aligned line segments in $$O\left( n\log n\right)$$ time, by a reduction to dynamic orthogonal point location.

## Keywords

Shortest paths Geometric intersection graphs Intersection searching data structures Disk graphs

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