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 $O(n\log2 n\log\log n)$ 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.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.