Spanners of Additively Weighted Point Sets

TL;DR

This paper investigates geometric spanners for additively weighted point sets, introducing a Yao graph variant with linear edges and analyzing the Additively Weighted Delaunay graph's spanning properties, including plane embeddings.

Contribution

It presents a new spanner construction for weighted points and analyzes the properties of Additively Weighted Delaunay graphs, including plane embeddings with constant spanning ratios.

Findings

Yao graph variant is a (1+ε)-spanner with linear edges for positive weights.

Additively Weighted Delaunay graph has constant spanning ratio.

A plane embedding of the Delaunay graph with constant spanning ratio can be computed.

Abstract

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs $(p,r)$ where $p$ is a point in the plane and $r$ is a real number. The distance between two points $(p_i,r_i)$ and $(p_j,r_j)$ is defined as $|p_ip_j|-r_i-r_j$. We show that in the case where all $r_i$ are positive numbers and $|p_ip_j|\geq r_i+r_j$ for all $i,j$ (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a $(1+\epsilon)$-spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has constant spanning ratio. The straight line embedding of the Additively Weighted Delaunay graph may not be a plane graph. We show how to compute a plane embedding that also has a constant spanning ratio.