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Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.
Let $mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_mathcal{D}$ for $mathcal{D}$ is the undirected graph with vertex set $mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission
Let $P subset mathbb{R}^2$ be a planar $n$-point set such that each point $p in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q in P$, there is an edge from $
For any constants $dge 1$, $epsilon >0$, $t>1$, and any $n$-point set $Psubsetmathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(nlog^2 nloglog n)$ edges with the following property: For any $Fsubseteq P$, there exists $F^+sups
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
A conflict-free $k$-coloring of a graph $G=(V,E)$ assigns one of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among $v$ and $v$s neighbors. Such colorings have a