New constructions of SSPDs and their applications
Proceedings of the twenty-sixth annual symposium on Computational geometry
The euclidean bottleneck steiner path problem
Proceedings of the twenty-seventh annual symposium on Computational geometry
New constructions of SSPDs and their applications
Computational Geometry: Theory and Applications
On the power of the semi-separated pair decomposition
Computational Geometry: Theory and Applications
Proceedings of the twenty-ninth annual symposium on Computational geometry
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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 $\mathcal{G}$on a point set P and a region F, we define $\mathcal{G}\ominus F$to be what remains of $\mathcal{G}$after the vertices and edges of $\mathcal{G}$intersecting F have been removed. A $\mathcal{C}$-fault tolerant t-spanner is a geometric graph $\mathcal{G}$on P such that for any convex region F, the graph $\mathcal{G}\ominus F$is a t-spanner for $\mathcal{G}_{c}(P)\ominus F$, where $\mathcal{G}_{c}(P)$is the complete geometric graph on P. We prove that any set P of n points admits a $\mathcal{C}$-fault tolerant (1+ε)-spanner of size $\mathcal{O}(n\log n)$for any constant ε0; if adding Steiner points is allowed, then the size of the spanner reduces to $\mathcal{O}(n)$, and for several special cases, we show how to obtain region-fault tolerant spanners of $\mathcal{O}(n)$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 $\mathcal{G}\ominus D$between any two points u,v∈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 $\mathcal{O}(n)$Steiner points to obtain a fault-tolerant geodesic (1+ε)-spanner of size $\mathcal{O}(n)$.