Dynamic Connectivity in Disk Graphs

Abstract

Let S subseteq mathbb {R}^2 be a set of nsites in the plane, so that every site s in S has an associated radiusr_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 ratioPsi , 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(nlog Psi log ^{4} n) overall expected time for disk graphs with bounded radius ratio Psi and O(nlog ^{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.