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Let $S$ be a set of $n$ sites, each associated with a point in $mathbb{R}^2$ and a radius $r_s$ and let $mathcal{D}(S)$ be the disk graph on $S$. We consider the problem of designing data structures that maintain the connectivity structure of $mathcal{D}(S)$ 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 $Psi$ 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 $O(Psi lambda_6(log n) log^{9}n)$ update time and $O(log n)$ query time, where $lambda_s(n)$ 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 $Psi$ at the cost of an additional $O(log log n)$ factor in the query time. In the incremental case we manage to achieve a logarithmic dependency on $Psi$ with a data structure with $O(alpha(n))$ query and $O(logPsi lambda_6(log n) log^{9}n)$ 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 $O((log n)/(log log n))$ supporting $m$ deletions in $O((nlog^{5}n + m log^{9}n) lambda_6(log n) + nlogPsilog^4n)$ overall time for bounded radius ratio $Psi$ and $O(( nlog^{6} n + m log^{10}n) lambda_6(log n))$ for general disk graphs.
Let $Vsubsetmathbb{R}^2$ be a set of $n$ sites in the plane. The unit disk graph $DG(V)$ of $V$ is the graph with vertex set $V$ in which two sites $v$ and $w$ are adjacent if and only if their Euclidean distance is at most $1$. We develop a compact
Given a set P of n points in the plane, a unit-disk graph G_{r}(P) with respect to a radius r is an undirected graph whose vertex set is P such that an edge connects two points p, q in P if the Euclidean distance between p and q is at most r. The len
Let $S subset mathbb{R}^2$ be a set of $n$ sites, where each $s in S$ has an associated radius $r_s > 0$. The disk graph $D(S)$ is the undirected graph with vertex set $S$ and an undirected edge between two sites $s, t in S$ if and only if $|st| leq
Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $mathcal{O}(log^*n)$-round distributed algorithm that given
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark 90, Raghavan and Spinrad 03] to translates of any fixed convex set