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 Communicated by Joseph S.B. Mitchell. M.A. Discrete Comput Geom (2009) 41: 556-582 557remains 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 G c (P ) F , where G c (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(n log 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 R 2 \ D. 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).

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The greedy algorithm produces high-quality spanners and, therefore, is used in several applications. However, even for points in d-dimensional Euclidean space, the greedy algorithm has near-cubic running time. In this paper, we present an algorithm that computes the greedy spanner for a set of n points in a metric space with bounded doubling dimension in O(n 2 log n) time. Since computing the greedy spanner has an Ω(n 2 ) lower bound, the time complexity of our algorithm is optimal within a logarithmic factor.