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 Gc(P)⊖F, where Gc(P) is the complete geometric graph on P. We prove that any set P of n points admits a C-fault tolerant (1 + e)-spanner of size O(n log n), for any constant e > 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 e P D is at most t times the geodesic distance between u and v in R2 D. We prove that for any P we can add O(n) Steiner points to obtain a fault-tolerant geodesic (1 + e)-spanner of size O(n).