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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

A ν-reliable spanner of a metric space (X,d), is a (dominating) graph H, such that for any possible failure set B ⊆ X, there is a set B^+ just slightly larger |B^+| ≤ (1+ν)⋅|B|, and all distances between pairs in X⧵B^+ are (approximately) preserved in H⧵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 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 oblivious reliable spanner is a distribution over spanners, and the bound on |B^+| holds in expectation.
We devise an oblivious ν-reliable (2+2/(k-1))-spanner for any k-HST, whose lightness is ≈ 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 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+ε)-spanner with lightness ε^{-O(ddim)} ⋅ Õ(ν^{-2}⋅log n), which is best possible (up to lower order terms).

Arnold Filtser, Yuval Gitlitz, and Ofer Neiman. Light, Reliable Spanners. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{filtser_et_al:LIPIcs.SoCG.2024.56, author = {Filtser, Arnold and Gitlitz, Yuval and Neiman, Ofer}, title = {{Light, Reliable Spanners}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {56:1--56:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.56}, URN = {urn:nbn:de:0030-drops-200019}, doi = {10.4230/LIPIcs.SoCG.2024.56}, annote = {Keywords: light spanner, reliable spanner, HST cover, doubling metric, minor free graphs} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Let G = (V,E,w) be a weighted undirected graph with n vertices and m edges, and fix a set of s sources S ⊆ V. We study the problem of computing almost shortest paths (ASP) for all pairs in S × V in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of 1+ε, for an arbitrarily small constant ε > 0 (henceforth (1+ε)-ASP for S × V). In this regime existing centralized algorithms require Ω(min{|E|s,n^ω}) time, where ω < 2.372 is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work Ω(min{|E|s,n^ω}).
In a bold attempt to achieve centralized time close to the lower bound of m + n s, Cohen [Edith Cohen, 2000] devised an algorithm which, in addition to the multiplicative stretch of 1+ε, allows also additive error of β ⋅ W_{max}, where W_{max} is the maximum edge weight in G (assuming that the minimum edge weight is 1), and β = (log n)^{O((log 1/ρ)/ρ)} is polylogarithmic in n. It also depends on the (possibly) arbitrarily small parameter ρ > 0 that determines the running time O((m + ns)n^ρ) of the algorithm.
The tradeoff of [Edith Cohen, 2000] was improved in [M. Elkin, 2001], whose algorithm has similar approximation guarantee and running time, but its β is (1/ρ)^{O((log 1/ρ)/ρ)}. However, the latter algorithm produces distance estimates rather than actual approximate shortest paths. Also, the additive terms in [Edith Cohen, 2000; M. Elkin, 2001] depend linearly on a possibly quite large global maximum edge weight W_{max}.
In the current paper we significantly improve this state of affairs. Our centralized algorithm has running time O((m+ ns)n^ρ), and its PRAM counterpart has polylogarithmic depth and work O((m + ns)n^ρ), for an arbitrarily small constant ρ > 0. For a pair (s,v) ∈ S× V, it provides a path of length d̂(s,v) that satisfies d̂(s,v) ≤ (1+ε)d_G(s,v) + β ⋅ W(s,v), where W(s,v) is the weight of the heaviest edge on some shortest s-v path. Hence our additive term depends linearly on a local maximum edge weight, as opposed to the global maximum edge weight in [Edith Cohen, 2000; M. Elkin, 2001]. Finally, our β = (1/ρ)^{O(1/ρ)}, i.e., it is significantly smaller than in [Edith Cohen, 2000; M. Elkin, 2001].
We also extend a centralized algorithm of Dor et al. [D. Dor et al., 2000]. For a parameter κ = 1,2,…, this algorithm provides for unweighted graphs a purely additive approximation of 2(κ -1) for all pairs shortest paths (APASP) in time Õ(n^{2+1/κ}). Within the same running time, our algorithm for weighted graphs provides a purely additive error of 2(κ - 1) W(u,v), for every vertex pair (u,v) ∈ binom(V,2), with W(u,v) defined as above.
On the way to these results we devise a suite of novel constructions of spanners, emulators and hopsets.

Michael Elkin, Yuval Gitlitz, and Ofer Neiman. Almost Shortest Paths with Near-Additive Error in Weighted Graphs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{elkin_et_al:LIPIcs.SWAT.2022.23, author = {Elkin, Michael and Gitlitz, Yuval and Neiman, Ofer}, title = {{Almost Shortest Paths with Near-Additive Error in Weighted Graphs}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {23:1--23:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.23}, URN = {urn:nbn:de:0030-drops-161833}, doi = {10.4230/LIPIcs.SWAT.2022.23}, annote = {Keywords: spanners, hopset, shortest paths, PRAM, distance oracles} }

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**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

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 [Abu Reyan Ahmed et al., 2020] 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, [Abu Reyan Ahmed et al., 2020] 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+ε)W spanner for weighted graphs with size O(n^{4/3}) (for any constant ε > 0), thus nearly matching the classical +6 spanner of size O(n^{4/3}) for unweighted graphs. Furthermore, we show a +(2+ε)W subsetwise spanner of size O(n⋅√{|S|}), improving the +4W_{max} result of [Abu Reyan Ahmed et al., 2020] (that had the same size). We also show a simple randomized algorithm for a +4W emulator of size Õ(n^{4/3}).
In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. It is known that such spanners must suffer polynomially large stretch. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size +Õ(√n⋅ W) spanner, and we also obtain a tradeoff between size and stretch.
Finally, generalizing the technique of [D. Dor et al., 2000] for unweighted graphs, we devise an efficient randomized algorithm producing a +2W spanner for weighted graphs of size Õ(n^{3/2}) in Õ(n²) time.

Michael Elkin, Yuval Gitlitz, and Ofer Neiman. Improved Weighted Additive Spanners. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{elkin_et_al:LIPIcs.DISC.2021.21, author = {Elkin, Michael and Gitlitz, Yuval and Neiman, Ofer}, title = {{Improved Weighted Additive Spanners}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.21}, URN = {urn:nbn:de:0030-drops-148232}, doi = {10.4230/LIPIcs.DISC.2021.21}, annote = {Keywords: Graph theory, Pure additive spanners} }

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