All-pairs shortest paths in geometric intersection graphs

Abstract

$\newcommand{\OO}[1]{O\left(#1\right)}$We present a simple and general algorithm for the all-pairs shortest paths (APSP) problem in unweighted geometric intersection graphs. Specifically we reduce the problem to the design of static data structures for offline intersection detection. Consequently we can solve APSP in unweighted intersection graphs of $n$ arbitrary disks in $\OO{n^2 \log n}$ time, axis-aligned line segments in $\OO{n^2 \log{\log n}}$ time, arbitrary line segments in $\OO{n^{7/3} \log^{1/3} n}$ time, $d$-dimensional axis-aligned unit hypercubes in $\OO{n^2 \log\log n}$ time for $d=3$ and $\OO{n^2 \log^{d-3} n}$ time for $d\geq4$, and $d$-dimensional axis-aligned boxes in $\OO{n^2 \log^{d-1.5} n}$ time for $d\geq2$. We also reduce the single-source shortest paths (SSSP) problem in unweighted geometric intersection graphs to decremental intersection detection. Thus, we obtain an $\OO{n \log n}$-time SSSP algorithm in unweighted intersection graphs of $n$ axis-aligned line segments.