A \emph{$\nu$-reliable spanner} of a metric space $(X,d)$, is a (dominating) graph $H$, such that for any possible failure set $B\subseteq X$, there is a set $B^+$ just slightly larger $|B^+|\le(1+\nu)\cdot|B|$, and all distances between pairs in $X\setminus B^+$ are (approximately) preserved in $H\setminus B$. Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all. In this work, we initiate the study of \emph{light} reliable spanners, whose weight is proportional to that of the Minimum Spanning Tree (MST) of $X$. We first observe that unlike sparsity, the lightness of any deterministic reliable spanner is huge, even for the metric of the simple path graph. Therefore, randomness must be used: an \emph{oblivious} reliable spanner is a distribution over spanners, and the bound on $|B^+|$ holds in expectation. We devise an oblivious $\nu$-reliable $(2+\frac{2}{k-1})$-spanner for any $k$-HST, whose lightness is $\approx \nu^{-2}$. We demonstrate a matching $\Omega(\nu^{-2})$ lower bound on the lightness (for any finite stretch). We also note that any stretch below 2 must incur linear lightness. For general metrics, doubling metrics, and metrics arising from minor-free graphs, we construct {\em light} tree covers, in which every tree is a $k$-HST of low weight. Combining these covers with our results for $k$-HSTs, we obtain oblivious reliable light spanners for these metric spaces, with nearly optimal parameters. In particular, for doubling metrics we get an oblivious $\nu$-reliable $(1+\varepsilon)$-spanner with lightness $\varepsilon^{-O({\rm ddim})}\cdot\tilde{O}(\nu^{-2}\cdot\log n)$, which is best possible (up to lower order terms).

翻译：度量空间$(X,d)$的$\nu$-可靠扩展图是一个（支配性）图$H$，使得对于任意可能的故障集$B\subseteq X$，存在一个略大于原集的集合$B^+$满足$|B^+|\le(1+\nu)\cdot|B|$，并且$X\setminus B^+$中所有点对之间的距离在$H\setminus B$中（近似）保持不变。近年来，虽有多项研究在不同场景下构造稀疏的可靠扩展图，但此类图的权重问题始终未被分析。本文首次研究\textit{轻量}可靠扩展图，其权重与$X$的最小生成树(MST)权重成比例。我们首先发现，即便对于简单路径图度量，任何确定性可靠扩展图的轻量性都难以保证（远弱于稀疏性），因此必须引入随机性：\textit{无感知}可靠扩展图是扩展图上的一个概率分布，且$|B^+|$的界限在期望意义下成立。针对任意$k$-HST，我们设计了一种无感知$\nu$-可靠$(2+\frac{2}{k-1})$-扩展图，其轻量性指标约为$\nu^{-2}$，并证明了轻量性下界$\Omega(\nu^{-2})$（对任意有限伸缩比均成立）。我们还指出，任何伸缩比低于2的扩展图必然具有线性轻量性。对于一般度量、加倍度量以及由无小禁子图导出的度量，我们构造了\textit{轻量}树覆盖，其中每棵树均为低权重的$k$-HST。将这些覆盖与$k$-HST的结果结合，我们获得了这些度量空间上近最优参数的无感知可靠轻量扩展图。特别地，对于加倍度量，我们得到一种无感知$\nu$-可靠$(1+\varepsilon)$-扩展图，其轻量性指标为$\varepsilon^{-O({\rm ddim})}\cdot\tilde{O}(\nu^{-2}\cdot\log n)$，该结果在（低阶项意义下）达到最优。