Spanners of additively weighted point sets

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs (p,r)(p,r) where p is a point in the plane and r   is a real number. The distance between two points (pi,ri)(pi,ri) and (pj,rj)(pj,rj) is defined as |pipj|−ri−rj|pipj|−ri−rj. We show that in the case where all riri are positive numbers and |pipj|⩾ri+rj|pipj|⩾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)(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.