Local Geometric Spanners

Abstract

We introduce the concept of local spanners for planar point sets with respect to a family of regions, and prove the existence of local spanners of small size for some families. For a geometric graph G on a point set $P$ and a region R belonging to a family ${\mathcal {R}}$, we define $G \cap R$ to be the part of the graph G that is inside R (or is induced by R). A local t-spanner w.r.t ${\mathcal {R}}$ is a geometric graph G on $P$ such that for any region $R \in {\mathcal {R}}$, the graph $G\cap R$ is a t-spanner for $K(P) \cap R$, where $K(P)$ is the complete geometric graph on P. For any set $P$ of n points and any constant $\varepsilon > 0$, we prove that $P$ admits local $(1 + \varepsilon )$-spanners of sizes $O(n \log ^{6} n)$ and $O(n \log n)$ w.r.t axis-parallel squares and vertical slabs, respectively. If adding Steiner points is allowed, then local $(1 + \varepsilon )$-spanners with O(n) edges and $O(n \log ^2 n)$ edges can be obtained for axis-parallel squares and disks using O(n) Steiner points, respectively.