On generalized diamond spanners

Abstract

Given a set P of points in the plane and a set L of non-crossing line segments whose endpoints are in P, a constrained plane geometric graph is a plane graph whose vertex set is P and whose edge set contains L. An edge e has the α-visible diamond property if one of the two isosceles triangles with base e and base angle a does not contain any points of P visible to both endpoints of e. A constrained plane geometric graph has the d-good polygon property provided that for every pair x, y of visible vertices on a face /, the shorter of the two paths from x to y around the boundary has length at most d ·|xy|. If a constrained plane geometric graph has the a-visible diamond property for each of its edges and the d-good polygon property, we show it is a 8d(π-α)2/α2sin2(α/4)- spanner of the visibility graph of P and L. This is a generalization of the result by Das and Joseph [3] to the constrained setting as well as a slight improvement on their spanning ratio of 8dπ) 2/α2sin2(α/4). We then show that several well-known constrained triangulations (namely the constrained Delaunay triangulation, constrained greedy triangulation and constrained minimum weight triangulation) have the α-visible diamond property for some constant α. In particular, we show that the greedy triangulation has the π/6-visible diamond property, which is an improvement over previous results. © Springer-Verlag Berlin Heidelberg 2007.