  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: failure of even $90\%$ 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.
