Dynamic Connectivity in Disk Graphs

Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated radius rs > 0. Let 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 ∈ 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 in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S. For this case, we describe a data structure that has O(log2 n) amortized update time and O(log n/ log log n) 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 parameter Ψ 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(Ψ log4 n) and query time O(log n/ log log n). This improves the currently best update time by a factor of Ψ. In the incremental case, we achieve logarithmic dependency on Ψ, with a data structure that has O(α(n)) amortized query time and O(log Ψ log4 n) amortized expected update time, where α(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 Ψ log4 n) overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 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

Planar Hop Spanners for Unit Disk Graphs

International audienceThe simplest modelof a wireless network graph is the Unit Disk Graph (UDG): an edgeexists in UDG if the Euclidean distance between its endpoints is$\le 1$. The problem of constructing planar spanners of Unit DiskGraphs with respect to the Euclidean distance has receivedconsiderable attention from researchers in computational geometryand ad-hoc wireless networks. In this paper, we present an algorithmthat, given a set $X$ of terminals in the plane, constructs a planarhop spanner with constant stretch factor for the Unit Disk Graphdefined by $X$. Our algorithm improves on previous constructions inthe sense that (i) it ensures the planarity of the whole spannerwhile previous algorithms ensure only the planarity of a backbonesubgraph; (ii) the hop stretch factor of our spanner issignificantly smaller