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:
We 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 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.
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M.A.A. was supported by Technische Universität Dortmund and the MADALGO Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation.
M.d.B. was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.
M.F. and M.S. were supported by Natural Sciences and Engineering Research Council (NSERC) of Canada.
NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.
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Abam, M.A., de Berg, M., Farshi, M. et al. Geometric Spanners for Weighted Point Sets. Algorithmica 61, 207–225 (2011). https://doi.org/10.1007/s00453-010-9465-2
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DOI: https://doi.org/10.1007/s00453-010-9465-2