A \emph{$ν$-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 $|B^+|\le(1+ν)\cdot|B|$, 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 $ν$-reliable $(2+\frac{2}{k-1})$-spanner for any $k$-HST, whose lightness is $\approx ν^{-2}$. We demonstrate a matching $Ω(ν^{-2})$ 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 $ν$-reliable $(1+\varepsilon)$-spanner with lightness $\varepsilon^{-O({\rm ddim})}\cdot\tilde{O}(ν^{-2}\cdot\log n)$, which is best possible (up to lower order terms).

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). extended the classical +2 (respectively, +4) spanners for unweighted graphs of size $$O(n^{3/2})$$ (resp., $$O(n^{7/5})$$ ) to the weighted setting, where the additive error is $$+2W$$ (resp., $$+4W$$ ). This means that for every pair u, v, the additive stretch is at most $$+2W_{u,v}$$ , where $$W_{u,v}$$ is the maximal edge weight on the shortest $$u-v$$ path (weights are normalized so that the minimum edge weight is 1). In addition, as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). showed a randomized algorithm yielding a $$+8W_{max}$$ spanner of size $$O(n^{4/3})$$ , here $$W_{max}$$ is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a $$+(6+\varepsilon )W$$ spanner for weighted graphs with size $$O(n^{4/3})$$ (for any constant $$\varepsilon >0$$ ), thus nearly matching the classical +6 spanner of size $$O(n^{4/3})$$ for unweighted graphs. Furthermore, we show a $$+(2+\varepsilon )W$$ subsetwise spanner of size $$O(n\cdot \sqrt{\vert S\vert })$$ , improving the $$+4W_{max}$$ result of as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). (that had the same size). We also show a simple randomized algorithm for a $$+4W$$ emulator of size $${\tilde{O}}(n^{4/3})$$ . In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. It was proved by Abboud A, Bodwin G (J ACM 64(4):28–12820 2017) that such spanners must suffer polynomially large stretches. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size $$+{\tilde{O}}(\sqrt{n}\cdot W)$$ spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of Dor D et al. (SIAM J Comput 29:1740–1759, 2000) for unweighted graphs, we devise an efficient randomized algorithm producing a $$+2W$$ spanner for weighted graphs of size $${\tilde{O}}(n^{3/2})$$ in $${\tilde{O}}(n^2)$$ time.