Region-fault tolerant geometric spanners

Abstract

We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph 𝒢 on a point set P and a region F, we define 𝒢 ⊖ F to be what remains of 𝒢 after the vertices and edges of 𝒢 intersecting F have been removed. A C-fault tolerant t-spanner is a geometric graph 𝒢 on P such that for any convex region F, the graph 𝒢 ⊖ F is a t-spanner for 𝒢Gc(P) ⊖ F , where 𝒢Gc(P) is the complete geometric graph on P. We prove that any set P of n points admits a C-fault tolerant (1+ε)-spanner of size O (nlog n) for any constant ε>0; if adding Steiner points is allowed, then the size of the spanner reduces to O (n), and for several special cases, we show how to obtain region-fault tolerant spanners of O(n) size without using Steiner points. We also consider fault-tolerant geodesic t -spanners: this is a variant where, for any disk D, the distance in 𝒢 ⊖ D between any two points u,v P D is at most t times the geodesic distance between u and v in ℝ2D. We prove that for any P, we can add O(n) Steiner points to obtain a fault-tolerant geodesic (1+ε)-spanner of size O (n). © 2009 Springer Science+Business Media, LLC.