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

TL;DR

This paper constructs sparse geometric graphs called robust spanners that maintain approximate shortest paths even after removing a small subset of points, with fewer edges than previous methods, for any fixed dimension and parameters.

Contribution

It introduces a new construction of $(1+psilon)k$-robust $t$-spanners with $O(n\u200b ext{log}^2 n\u200b ext{log}\u200b ext{log} n)$ edges, improving sparsity over prior work.

Findings

Constructed a $(1+psilon)k$-robust $t$-spanner with $O(n\u200b ext{log}^2 n\u200b ext{log}\u200b ext{log} n)$ edges.

The spanner maintains approximate shortest paths after removing a small subset of points.

The construction is sparser than recent existing robust 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, Ol\`ah, and Har-Peled (arXiv:1811.06898) who prove the existence of $(1+\epsilon)k$-robust $t$-spanners with $n\log^{O(d)} n$ edges.