Geometric Spanners for Weighted Point Sets
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 weighted distance function d w , where d w (p,q) is w(p) + d(p,q) + wq if p ≠ q and 0 otherwise. We present a general method for turning spanners with respect to the d-metric into spanners with respect to the d w -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 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(nlogn). 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.
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