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 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+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.