Approximate Distance Queries in Disk Graphs

Abstract

We present efficient algorithms for approximately answering distance queries in disk graphs. Let G be a disk graph with n vertices and m edges. For any fixed ε> 0, we show that G can be preprocessed in $O(m\sqrt{n}\epsilon^{-1}+m\epsilon^{-2}\log S)$ time, constructing a data structure of size O(n ^3/2 ε ^− 1+nε ^− 2logS), such that any subsequent distance query can be answered approximately in $O(\sqrt{n}\epsilon^{-1}+\epsilon^{-2}\log S)$ time. Here S is the ratio between the largest and smallest radius. The estimate produced is within an additive error which is only ε times the longest edge on some shortest path.

The algorithm uses an efficient subdivision of the plane to construct a sparse graph having many of the same distance properties as the input disk graph. Additionally, the sparse graph has a small separator decomposition, which is then used to answer distance queries. The algorithm extends naturally to the higher dimensional ball graphs.