Robust geometric spanners

Abstract

Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.

A Spanner for the Day After

“…It becomes significantly more challenging to construct such an exact spanner that is reliable. In case of robust spanners, for a function f under some general conditions, Bose et al [9] gave a construction of exact f (k)-robust spanners. In particular, their result implies that one can construct O(k log k)-robust 1-spanners of size O(n log n) and, for any ε > 0, O(k 1+ε )-robust 1-spanners of size O(n log log n), for one-dimensional point sets.…”

“…Robustness Here, our goal is to construct spanners that are robust according to the notion introduced by Bose et al [9]. Intuitively, a spanner is robust if the deletion of k vertices only harms a few other vertices.…”

“…A priori it is not clear that such a sparse graph should exist (for t a constant) for a point set in R d , since the robustness property looks quite strong. Surprisingly, Bose et al [9] showed that one can construct a O(k 2 )-robust O(1)-spanner with O(n log n) edges. Bose et al [9] proved various other bounds in the same vein on the size of one-dimensional and higher-dimensional point sets.…”

“…Surprisingly, Bose et al [9] showed that one can construct a O(k 2 )-robust O(1)-spanner with O(n log n) edges. Bose et al [9] proved various other bounds in the same vein on the size of one-dimensional and higher-dimensional point sets. Their most closely related result is that for the one-dimensional point set, P = {1, 2, .…”

“…An open problem left by Bose et al [9] is the construction of O(k)-robust spannersthey only provide the easy upper bound of O(n 2 ) for this case. In this paper, we present several constructions for this case with optimal or near-optimal size.…”