Near-Optimal $O(k)$-Robust Geometric Spanners

Abstract

For any constants $d\ge 1$, $\epsilon >0$, $t>1$, and any $n$-point set $P\subset {\mathbb{R}}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n{\log }^{2}n\log \log n)$ edges with the following property: For any $F\subseteq P$, there exists $F^{+}\supseteq F$, $|F^{+}|\le (1+\epsilon )|F|$ such that, for any pair $p,q\in P\setminus F^{+}$, the graph $G-F$ contains a path from $p$ to $q$ whose (Euclidean) length is at most $t$ times the Euclidean distance between $p$ and $q$. In the terminology of robust spanners (Bose \et al, SICOMP, 42(4):1720--1736, 2013) the graph $G$ is a $(1+\epsilon )k$-robust $t$-spanner of $P$. This construction is sparser than the recent constructions of Buchin, Olàh, and Har-Peled (arXiv:1811.06898) who prove the existence of $(1+\epsilon )k$-robust $t$-spanners with $n{\log }^{O(d)}n$ edges.

New version with streamlined construction and fewer edges