Robust Geometric Spanners

TL;DR

This paper introduces the concept of robustness in geometric spanners, demonstrating that while robust spanners require superlinear edges in one dimension, they can be constructed with near-linear edges in higher dimensions.

Contribution

It formally defines robustness for geometric spanners, proves superlinear edge requirements in one dimension, and provides near-linear constructions in higher dimensions.

Findings

Robust spanners require superlinear edges in one dimension.

Near-linear size robust spanners exist in higher dimensions.

Robustness limits and possibilities are characterized.

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.