All-Pairs Shortest Paths in Geometric Intersection Graphs

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