Spanners of Additively Weighted Point Sets

Authors:
Prosenjit Bose;Paz Carmi;Mathieu Couture
Affiliations:
School of Computer Science, Carleton University, Ottawa, Canada;School of Computer Science, Carleton University, Ottawa, Canada;School of Computer Science, Carleton University, Ottawa, Canada
Venue:
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Year:
2008

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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 pis a point in the plane and ris a real number. The distance between two points (pi,ri) and (pj,rj) is defined as |pipj| 茂戮驴 ri茂戮驴 rj. We show that in the case where all riare positive numbers and |pipj| 茂戮驴 ri+ rjfor 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 + 茂戮驴)-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. Given the Additively Weighted Delaunay graph, we show how to compute a plane embedding with a constant spanning ratio in O(nlogn) time.