A highly desirable property of networks is robustness to failures. Consider a metric space (X, d_X ]  ), a graph H over X is a \vartheta-reliable t-spanner if, for every set of failed vertices B\subset X, there is a superset B^+\supseteq B such that the induced subgraph H }} [ X\setminus B ] preserves all the distances between points in X\setminus B^+ up to a stretch factor t, while the expected size of B^+ is as most (1+\vartheta)|B|. Such a spanner could withstand a catastrophe: failureof even{\em    90\\%DIF < of the network. Buchin, Har-Peled, and Ol{\'{a}}h [2019,2020], constructed very sparse reliable spanners with stretch 1+\epsilon for Euclidean space using locality-sensitive orderings. Har-Peled and Ol{\'{a}}h [2020] constructed reliable spanners for various non-Euclidean metric spaces using sparse covers. However, this second approach has an inherent dependency on the aspect ratio (a.k.a. spread) and gives sub-optimal stretch and sparsity parameters. Our contribution is twofold: 1) We construct a locality-sensitive ordering for doubling metrics with a small number of orderings. As a corollary, we obtain reliable spanners for doubling metrics matching the sparsity parameters of known reliable spanners for Euclidean space. 2) We introduce new types of locality-sensitive orderings suitable for non-Euclidean metrics and construct such orderings for various metric families. We then construct reliable spanners from the newly introduced locality-sensitive orderings via reliable 2-hop spanners for paths. The number of edges in our spanner has no dependency on the spread.\end{abstract} %DIF > of the nodes fail. In a follow-up work, Har-Peled, Mendel, and Ol{\'{a}}h [2021] constructed reliable spanners for general and topologically structured metrics. Their construction used a different approach, and is based on sparse covers. In this paper, we develop the theory of LSO's to non-Euclidean metrics by introducing new types of LSO's suitable for general and topologically structured metrics. We then construct such LSO's, as well as constructing considerably improved LSO's for doubling metrics. Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters. Most prominently, we construct \tilde{O}(n)-size reliable spanners for trees and planar graphs with the optimal stretch of 2. Along the way to the construction of LSO's and reliable spanners, we introduce and construct ultrametric covers, and construct 2-hop reliable spanners for the line.
Chan, Har-Peled, and Jones 2020] recently developed locality-sensitive ordering (LSO ), a new tool that allows one to reduce problems in the Euclidean space \mathbb{R \'{a}}h [ 2019,2020 ] used the LSO of Chan{\em et al. to construct very sparsereliable spanners for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90\\%DIF < of the network. Buchin, Har-Peled, and Ol{\'{a}}h [2019,2020], constructed very sparse reliable spanners with stretch 1+\epsilon for Euclidean space using locality-sensitive orderings. Har-Peled and Ol{\'{a}}h [2020] constructed reliable spanners for various non-Euclidean metric spaces using sparse covers. However, this second approach has an inherent dependency on the aspect ratio (a.k.a. spread) and gives sub-optimal stretch and sparsity parameters. Our contribution is twofold: 1) We construct a locality-sensitive ordering for doubling metrics with a small number of orderings. As a corollary, we obtain reliable spanners for doubling metrics matching the sparsity parameters of known reliable spanners for Euclidean space. 2) We introduce new types of locality-sensitive orderings suitable for non-Euclidean metrics and construct such orderings for various metric families. We then construct reliable spanners from the newly introduced locality-sensitive orderings via reliable 2-hop spanners for paths. The number of edges in our spanner has no dependency on the spread.\end{abstract} %DIF > of the nodes fail. In a follow-up work, Har-Peled, Mendel, and Ol{\'{a}}h [2021] constructed reliable spanners for general and topologically structured metrics. Their construction used a different approach, and is based on sparse covers. In this paper, we develop the theory of LSO's to non-Euclidean metrics by introducing new types of LSO's suitable for general and topologically structured metrics. We then construct such LSO's, as well as constructing considerably improved LSO's for doubling metrics. Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters. Most prominently, we construct \tilde{O}(n)-size reliable spanners for trees and planar graphs with the optimal stretch of 2. Along the way to the construction of LSO's and reliable spanners, we introduce and construct ultrametric covers, and construct 2-hop reliable spanners for the line.
R
A highly desirable property of networks is robustness to failures. Consider a metric space (X, d_X
Chan, Har-Peled, and Jones
A
2020
A
recently developed locality-sensitive ordering (LSO
R
graph H over X is a \vartheta-reliable t-spanner if, for every set of failed vertices B\subset X, there is a superset B^+\supseteq B such that the induced subgraph H
new tool that allows one to reduce problems in the Euclidean space \mathbb{R
A
\'{a
A
h
R
X\setminus B
2019,2020
R
preserves all the distances between points in X\setminus B^+ up to a stretch factor t, while the expected size of B^+ is as most (1+\vartheta)|B|. Such a spanner could withstand a catastrophe: failureof even
used the LSO of Chan
A
et al.
A
to construct very sparse
A
reliable spanners
A
for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if