$\mathcal{O}(k)$-robust spanners in one dimension

TL;DR

This paper constructs sparse, $ ext{O}(k)$-robust 1-spanners for one-dimensional point sets, significantly improving edge efficiency over previous dense constructions while maintaining robustness.

Contribution

It introduces a method to build $ ext{O}(k)$-robust 1-spanners with near-linear edges in one dimension, advancing the understanding of robust geometric spanners.

Findings

Existence of $ ext{O}(k)$-robust 1-spanners with $ ext{O}(n^{1+ ext{epsilon}})$ edges

Previous constructions required $ ext{O}(n^2)$ edges

Certain point sets require at least $ ext{Omega}(n ext{log} n)$ edges for robust spanners

Abstract

A geometric $t$-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most $t$ times the Euclidean distance between the points. Informally, a spanner is $\mathcal{O}(k)$-robust if deleting $k$ vertices only harms $\mathcal{O}(k)$ other vertices. We show that on any one-dimensional set of $n$ points, for any $\varepsilon>0$, there exists an $\mathcal{O}(k)$-robust $1$-spanner with $\mathcal{O}(n^{1+\varepsilon})$ edges. Previously it was only known that $\mathcal{O}(k)$-robust spanners with $\mathcal{O}(n^2)$ edges exists and that there are point sets on which any $\mathcal{O}(k)$-robust spanner has $\Omega(n\log{n})$ edges.