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) 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 |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 + )-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. ∗Research partially supported by NSERC, MRI, CFI, and MITACS. 1 ar X iv :0 80 1. 40 13 v1 [ cs .C G ] 2 5 Ja n 20 08