Geometric spanners for weighted point sets

Abstract

Let (S,d) be a finite metric space, where each element p S has a non-negative weight w (p). We study spanners for the set S with respect to the following weighted distance function: d ω(p,q)= 0 if p = Q, {w}(p)+ d(p,q)+ w(q) if p ≠ q qWe present a general method for turning spanners with respect to the d-metric into spanners with respect to the d ω -metric. For any given ε>0, we can apply our method to obtain (5+ε)-spanners with a linear number of edges for three cases: points in Euclidean space ℝ d, points in spaces of bounded doubling dimension, and points on the boundary of a convex body in ℝ d where d is the geodesic distance function. We also describe an alternative method that leads to (2+ε)-spanners for weighted point points in ℝ d and for points on the boundary of a convex body in ℝ d. The number of edges in these spanners is O(nlog n). This bound on the stretch factor is nearly optimal: in any finite metric space and for any ε>0, it is possible to assign weights to the elements such that any non-complete graph has stretch factor larger than 2-ε. © 2010 The Author(s).