Region-fault tolerant geometric spanners

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 + 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 mos...

• Type: Conference
• Year: 2007
• Where: SODA
• Authors: Mohammad Ali Abam, Mark de Berg, Mohammad Farshi, Joachim Gudmundsson
• Keywords: Algorithms, Geometric Graph, Point Set, Region-fault Tolerant Spanners, SODA 2007