Dynamic Connectivity in Disk Graphs

Let S ⊆ R 2 be a set of n planar sites, such that each s ∈ S has an associated radius rs < 0. Let D(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii rs, rt intersect. Our goal is to design data structures that maintain the connectivity structure of D(S) as sites are inserted and/or deleted. First, we consider unit disk graphs, i.e., rs = 1, for all s ∈ S. We describe a data structure that has O(log2 n) amortized update and O(log n/ log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψλ6(log n)log7 n) and query time O(log n/ log log n), where λs(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(log Ψλ6(log n)log7 n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/ log log n) that support m deletions in O((n log5 n + m log7 n)λ6(log n) + n log Ψ log4 n) overall time for bounded radius ratio Ψ and O((n log6 n + m log8 n)λ6(log n)) for arbitrary radii.