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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 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Given an undirected weighted graph, an (approximate) distance oracle is a data structure that can (approximately) answer distance queries. A Path-Reporting Distance Oracle, or PRDO, is a distance oracle that must also return a path between the queried vertices. Given a graph on n vertices and an integer parameter k ≥ 1, Thorup and Zwick [M. Thorup and U. Zwick, 2001] showed a PRDO with stretch 2k-1, size O(k⋅n^{1+1/k}) and query time O(k) (for the query time of PRDOs, we omit the time needed to report the path itself). Subsequent works [Mendel and Naor, 2007; Shiri Chechik, 2014; Shiri Chechik, 2015] improved the size to O(n^{1+1/k}) and the query time to O(1). However, these improvements produce distance oracles which are not path-reporting. Several other works [Michael Elkin et al., 2016; Michael Elkin and Seth Pettie, 2016] focused on small size PRDO for general graphs, but all known results on distance oracles with linear size suffer from polynomial stretch, polynomial query time, or not being path-reporting.
In this paper we devise the first linear size PRDO with poly-logarithmic stretch and low query time O(log log n). More generally, for any integer k ≥ 1, we obtain a PRDO with stretch at most O(k^4.82), size O(n^{1+1/k}), and query time O(log k). In addition, we can make the size of our PRDO as small as n+o(n), at the cost of increasing the query time to poly-logarithmic. For unweighted graphs, we improve the stretch to O(k²).
We also consider pairwise PRDO, which is a PRDO that is only required to answer queries from a given set of pairs P. An exact PRDO of size O(n+|P|²) and constant query time was provided in [Michael Elkin and Seth Pettie, 2016]. In this work we dramatically improve the size, at the cost of slightly increasing the stretch. Specifically, given any ε > 0, we devise a pairwise PRDO with stretch 1+ε, constant query time, and near optimal size n^o(1)⋅ (n+|P|).

Ofer Neiman and Idan Shabat. Path-Reporting Distance Oracles with Linear Size. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{neiman_et_al:LIPIcs.SWAT.2024.36, author = {Neiman, Ofer and Shabat, Idan}, title = {{Path-Reporting Distance Oracles with Linear Size}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {36:1--36:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.36}, URN = {urn:nbn:de:0030-drops-200760}, doi = {10.4230/LIPIcs.SWAT.2024.36}, annote = {Keywords: Graph Algorithms, Shortest Paths, Distance Oracles} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Given an undirected possibly weighted n-vertex graph G = (V,E) and a set 𝒫 ⊆ V² of pairs, a subgraph S = (V,E') is called a P-pairwise α-spanner of G, if for every pair (u,v) ∈ 𝒫 we have d_S(u,v) ≤ α⋅ d_G(u,v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E'|/|P|.
A surprising connection was recently discussed between the additive stretch of (1+ε,β)-spanners, to the hopbound of (1+ε,β)-hopsets. A long sequence of works showed that if the spanner/hopset has size ≈ n^{1+1/k} for some parameter k ≥ 1, then β≈(1/ε)^{log k}. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P|≈ n^{1+1/k}, then a P-pairwise (1+ε)-spanner must have size at least β⋅ |P| with β≈(1/ε)^{log k} (a near matching upper bound was recently shown in [Michael Elkin and Idan Shabat, 2023]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners.
We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P|≈ n^{1+1/k}, then the size overhead is β≈k/α.
A source-wise spanner is a special type of pairwise spanner, for which P = A×V for some A ⊆ V. A prioritized spanner is given also a ranking of the vertices V = (v₁,… ,v_n), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-the-art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n⋅(log log n)²), that has a constant stretch for any query that contains a vertex ranked among the first n^{1-δ} vertices (for any constant δ > 0). Such a result was known before only for non-path-reporting distance oracles.

Ofer Neiman and Idan Shabat. On the Size Overhead of Pairwise Spanners. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 83:1-83:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{neiman_et_al:LIPIcs.ITCS.2024.83, author = {Neiman, Ofer and Shabat, Idan}, title = {{On the Size Overhead of Pairwise Spanners}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {83:1--83:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.83}, URN = {urn:nbn:de:0030-drops-196110}, doi = {10.4230/LIPIcs.ITCS.2024.83}, annote = {Keywords: Graph Algorithms, Shortest Paths, Spanners} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Given an undirected graph G = (V,E), an (α,β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, d_G(u,v) ≤ d_{G∪H}^(β)(u,v) ≤ α⋅ d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α.
In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm.
In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^ε),O(k^{1-ε}))-hopset of size Õ(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ε < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α,β)-hopset of size O(n^{1+1/k}) must have α⋅β ≥ Ω(k).

Ofer Neiman and Idan Shabat. A Unified Framework for Hopsets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{neiman_et_al:LIPIcs.ESA.2022.81, author = {Neiman, Ofer and Shabat, Idan}, title = {{A Unified Framework for Hopsets}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {81:1--81:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.81}, URN = {urn:nbn:de:0030-drops-170192}, doi = {10.4230/LIPIcs.ESA.2022.81}, annote = {Keywords: Graph Algorithms, Shortest Paths, Hopsets} }

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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 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Consider an undirected weighted graph G = (V,E,w). We study the problem of computing (1+ε)-approximate shortest paths for S × V, for a subset S ⊆ V of |S| = n^r sources, for some 0 < r ≤ 1. We devise a significantly improved algorithm for this problem in the entire range of parameter r, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of r in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time Õ(|E| ⋅ n^{o(1)} + n^{ω(r)}), where n^{ω(r)} is the time required to multiply an n^r × n matrix by an n × n one. Our PRAM algorithm has polylogarithmic time (log n)^{O(1/ρ)}, and its work complexity is Õ(|E| ⋅ n^ρ + n^{ω(r)}), for any arbitrarily small constant ρ > 0.
In particular, for r ≤ 0.313…, our centralized algorithm computes S × V (1+ε)-approximate shortest paths in n^{2 + o(1)} time. Our PRAM polylogarithmic-time algorithm has work complexity O(|E| ⋅ n^ρ + n^{2+o(1)}), for any arbitrarily small constant ρ > 0. Previously existing solutions either require centralized time/parallel work of O(|E| ⋅ |S|) or provide much weaker approximation guarantees.
In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for |S| = n^r sources, for r ≤ 0.655, while previous state-of-the-art algorithms did so only for r ≤ 1/2. Moreover, it improves previous bounds for all r > 1/2. For unweighted graphs, the running time is improved further to poly(log log n) for r ≤ 0.655. Previously this running time was known for r ≤ 1/2.

Michael Elkin and Ofer Neiman. Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{elkin_et_al:LIPIcs.STACS.2022.27, author = {Elkin, Michael and Neiman, Ofer}, title = {{Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {27:1--27:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.27}, URN = {urn:nbn:de:0030-drops-158379}, doi = {10.4230/LIPIcs.STACS.2022.27}, annote = {Keywords: Shortest paths, matrix multiplication, hopsets} }

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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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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995].
In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

Yair Bartal, Nova Fandina, and Ofer Neiman. Covering Metric Spaces by Few Trees. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bartal_et_al:LIPIcs.ICALP.2019.20, author = {Bartal, Yair and Fandina, Nova and Neiman, Ofer}, title = {{Covering Metric Spaces by Few Trees}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {20:1--20:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.20}, URN = {urn:nbn:de:0030-drops-105967}, doi = {10.4230/LIPIcs.ICALP.2019.20}, annote = {Keywords: tree cover, Ramsey tree cover, probabilistic hierarchical family} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n^{1+1/t^2}) edges, little is known.
In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of l_p with 1<p <=2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}.
In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Arnold Filtser and Ofer Neiman. Light Spanners for High Dimensional Norms via Stochastic Decompositions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filtser_et_al:LIPIcs.ESA.2018.29, author = {Filtser, Arnold and Neiman, Ofer}, title = {{Light Spanners for High Dimensional Norms via Stochastic Decompositions}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.29}, URN = {urn:nbn:de:0030-drops-94922}, doi = {10.4230/LIPIcs.ESA.2018.29}, annote = {Keywords: Spanners, Stochastic Decompositions, High Dimensional Euclidean Space, Doubling Dimension, Genus Graphs} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Given a metric space (X,d), a set of terminals K subseteq X, and a parameter t >= 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in K x X up to a factor of t, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, i.e., t=1+epsilon for some small 0<epsilon<1, is currently known.
Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1+epsilon and size s(|X|) has its terminal counterpart, with distortion 1+O(epsilon) and size s(|K|)+1. In particular, for any doubling metric on n points, a set of k=o(n) terminals, and constant 0<epsilon<1, there exists
- A spanner with stretch 1+epsilon for pairs in K x X, with n+o(n) edges.
- A labeling scheme with stretch 1+epsilon for pairs in K x X, with label size ~~ log k.
- An embedding into l_infty^d with distortion 1+epsilon for pairs in K x X, where d=O(log k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.

Michael Elkin and Ofer Neiman. Near Isometric Terminal Embeddings for Doubling Metrics. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{elkin_et_al:LIPIcs.SoCG.2018.36, author = {Elkin, Michael and Neiman, Ofer}, title = {{Near Isometric Terminal Embeddings for Doubling Metrics}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.36}, URN = {urn:nbn:de:0030-drops-87498}, doi = {10.4230/LIPIcs.SoCG.2018.36}, annote = {Keywords: metric embedding, spanners, doubling metrics} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric.
In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.

Alexandr Andoni, Assaf Naor, and Ofer Neiman. Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{andoni_et_al:LIPIcs.ICALP.2016.83, author = {Andoni, Alexandr and Naor, Assaf and Neiman, Ofer}, title = {{Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {83:1--83:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.83}, URN = {urn:nbn:de:0030-drops-62028}, doi = {10.4230/LIPIcs.ICALP.2016.83}, annote = {Keywords: Transportation metric, embedding, snowflake, sketching} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

In this paper we study terminal embeddings, in which one is given a finite metric (X,d_X) (or a graph G=(V,E)) and a subset K of X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve approx |K| * |X| pairs, the distortion depends only on |K|, rather than on |X|.
We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X \times X and with respect to K * X.
Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, Arora et. al. devised an ~O(sqrt(log(r))-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an ~O(sqrt(log |K|)-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K| <= r, our bound generalizes that of Arora et al.

Michael Elkin, Arnold Filtser, and Ofer Neiman. Terminal Embeddings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 242-264, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{elkin_et_al:LIPIcs.APPROX-RANDOM.2015.242, author = {Elkin, Michael and Filtser, Arnold and Neiman, Ofer}, title = {{Terminal Embeddings}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {242--264}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.242}, URN = {urn:nbn:de:0030-drops-53064}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.242}, annote = {Keywords: embedding, distortion, terminals} }

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