Dániel Oláh | Searcharxiv

arXiv2022

Reliable Spanners for Metric Spaces

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation, that is, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.

cs.CG↗

arXiv2018

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

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 $Ω(n\log{n})$ edges.

cs.CG↗