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 (pi,ri) and (pj,rj) is defined as |pipj|-ri-rj. We show that in the case where all ri are positive numbers and |pi pj|≥ri+rj 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+ε -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 a spanning ratio bounded by a constant. 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 straight-line embedding that also has a spanning ratio bounded by a constant in O(nlogn) time.

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Journal of Discrete Algorithms
School of Computer Science

Bose, P, Carmi, P. (Paz), & Couture, M. (Mathieu). (2011). Spanners of additively weighted point sets. In Journal of Discrete Algorithms (Vol. 9, pp. 287–298). doi:10.1016/j.jda.2011.03.001