Geodesic Spanners for Points on a Polyhedral Terrain

Let S be a set S of n points on a polyhedral terrain T in R, and let ε > 0 be a fixed constant. We prove that S admits a (2 + ε)-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in R admits an additively weighted (2 + ε)-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ε), and almost matches the lower bound. ∗Computer Engineering Department, Sharif University of Technology. Email: {abam,mjrezaei}@sharif.edu. †Department of Computer Science, TU Eindhoven, the Netherlands. Email: [email protected]. MdB was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 024.002.003. 1 ar X iv :1 51 1. 01 61 2v 1 [ cs .C G ] 5 N ov 2 01 5