All-pairs shortest paths in geometric intersection graphs

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 (n2 log n) time, axis-aligned line segments in O (n2 log log n) time, arbitrary line segments in O (n7/3 log1/3 n) time, d-dimensional axis-aligned boxes in O (n2 logd−1.5 n) time for d ≥ 2, and d-dimensional axis-aligned unit hypercubes in O (n2 log log n) time for d = 3 and O (n2 logd−3 n) time for d ≥ 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 (n log n) time, by a reduction to dynamic orthogonal point location.