Dynamic Connectivity in Disk Graphs

Abstract

Let \(S \subseteq \mathbb {R}^2\) be a set of n sites in the plane, so that every site \(s \in S\) has an associated radius \(r_s > 0\). Let \(\mathcal {D}(S)\) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites \(s, t \in S\) if and only if the disks with centers s, t and radii \(r_s\), \(r_t\) intersect. Our goal is to design data structures that maintain the connectivity structure of \(\mathcal {D}(S)\) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix \(r_s = 1\), for all sites \(s \in S\). For this case, we describe a data structure that has \(O(\log ^2 n)\) amortized update time and \(O(\log n/\log \log n)\) query time. Second, we look at disk graphs with bounded radius ratio \(\Psi \), i.e., for all \(s \in S\), we have \(1 \le r_s \le \Psi \), for a parameter \(\Psi \) that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time \(O(\Psi \log ^{4} n)\) and query time \(O(\log n/\log \log n)\). This improves the currently best update time by a factor of \(\Psi \). In the incremental case, we achieve logarithmic dependency on \(\Psi \), with a data structure that has \(O(\alpha (n))\) amortized query time and \(O(\log \Psi \log ^{4} n)\) amortized expected update time, where \(\alpha (n)\) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of \(O(\log n/\log \log n)\) that supports deletions in \(O(n\log \Psi \log ^{4} n)\) overall expected time for disk graphs with bounded radius ratio \(\Psi \) and \(O(n\log ^{5} n)\) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures.