Dynamic Connectivity in Disk Graphs
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Let S be a set of n sites, each associated with a point in ℝ^2 and a radius r_s and let 𝒟(S) be the disk graph on S. We consider the problem of designing data structures that maintain the connectivity structure of 𝒟(S) while allowing the insertion and deletion of sites. For unit disk graphs we describe a data structure that has O(log^2n) amortized update time and O((log n)/(loglog n)) amortized query time. For disk graphs where the ratio Ψ between the largest and smallest radius is bounded, we consider the decremental and the incremental case separately, in addition to the fully dynamic case. In the fully dynamic case we achieve amortized O(Ψλ_6(log n) log^9n) update time and O(log n) query time, where λ_s(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. This improves the update time of the currently best known data structure by a factor of Ψ at the cost of an additional O(loglog n) factor in the query time. In the incremental case we manage to achieve a logarithmic dependency on Ψ with a data structure with O(α(n)) query and O(logΨλ_6(log n) log^9n) update time. For the decremental setting we first develop a new dynamic data structure that allows us to maintain two sets B and P of disks, such than at a deletion of a disk from B we can efficiently report all disks in P that no longer intersect any disk of B. Having this data structure at hand, we get decremental data structures with an amortized query time of O((log n)/(loglog n)) supporting m deletions in O((nlog^5n + m log^9n) λ_6(log n) + nlogΨlog^4n) overall time for bounded radius ratio Ψ and O(( nlog^6 n + m log^10n) λ_6(log n)) for general disk graphs.
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