Geometric Spanners for Weighted Point Sets

AbstractLet (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: $$\mathbf{d}_{\omega}(p,q)=\left\{\begin{array}{ll}0&\mbox{ if $p=q$,}\\ \operatorname {w}(p)+\mathbf{d}(p,q)+ \operatorname {w}(q)&\mbox{ if $p\neq q$.}\end{array}\right.$$ We present a general method for turning spanners with respect to the d-metric into spanners with respect to the dω-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 ℝd, points in spaces of bounded doubling dimension, and points on the boundary of a convex body in ℝd where d is the geodesic distance function.We also describe an alternative method that leads to (2+ε)-spanners for weighted point points in ℝd and for points on the boundary of a convex body in ℝd. 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.