19 Search Results for "Filtser, Arnold"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.46
On the Streaming Complexity of Expander Decomposition

Authors: Yu Chen, Michael Kapralov, Mikhail Makarov, and Davide Mazzali

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In this paper we study the problem of finding (ε, ϕ)-expander decompositions of a graph in the streaming model, in particular for dynamic streams of edge insertions and deletions. The goal is to partition the vertex set so that every component induces a ϕ-expander, while the number of inter-cluster edges is only an ε fraction of the total volume. It was recently shown that there exists a simple algorithm to construct a (O(ϕ log n), ϕ)-expander decomposition of an n-vertex graph using Õ(n/ϕ²) bits of space [Filtser, Kapralov, Makarov, ITCS'23]. This result calls for understanding the extent to which a dependence in space on the sparsity parameter ϕ is inherent. We move towards answering this question on two fronts. We prove that a (O(ϕ log n), ϕ)-expander decomposition can be found using Õ(n) space, for every ϕ. At the core of our result is the first streaming algorithm for computing boundary-linked expander decompositions, a recently introduced strengthening of the classical notion [Goranci et al., SODA'21]. The key advantage is that a classical sparsifier [Fung et al., STOC'11], with size independent of ϕ, preserves the cuts inside the clusters of a boundary-linked expander decomposition within a multiplicative error. Notable algorithmic applications use sequences of expander decompositions, in particular one often repeatedly computes a decomposition of the subgraph induced by the inter-cluster edges (e.g., the seminal work of Spielman and Teng on spectral sparsifiers [Spielman, Teng, SIAM Journal of Computing 40(4)], or the recent maximum flow breakthrough [Chen et al., FOCS'22], among others). We prove that any streaming algorithm that computes a sequence of (O(ϕ log n), ϕ)-expander decompositions requires Ω̃(n/ϕ) bits of space, even in insertion only streams.
Cite as

Yu Chen, Michael Kapralov, Mikhail Makarov, and Davide Mazzali. On the Streaming Complexity of Expander Decomposition. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2024.46,
  author =	{Chen, Yu and Kapralov, Michael and Makarov, Mikhail and Mazzali, Davide},
  title =	{{On the Streaming Complexity of Expander Decomposition}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.46},
  URN =		{urn:nbn:de:0030-drops-201890},
  doi =		{10.4230/LIPIcs.ICALP.2024.46},
  annote =	{Keywords: Graph Sketching, Dynamic Streaming, Expander Decomposition}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.47
Lower Bounds on 0-Extension with Steiner Nodes

Authors: Yu Chen and Zihan Tan

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In the 0-Extension problem, we are given an edge-weighted graph G = (V,E,c), a set T ⊆ V of its vertices called terminals, and a semi-metric D over T, and the goal is to find an assignment f of each non-terminal vertex to a terminal, minimizing the sum, over all edges (u,v) ∈ E, the product of the edge weight c(u,v) and the distance D(f(u),f(v)) between the terminals that u,v are mapped to. Current best approximation algorithms on 0-Extension are based on rounding a linear programming relaxation called the semi-metric LP relaxation. The integrality gap of this LP, is upper bounded by O(log|T|/log log|T|) and lower bounded by Ω((log|T|)^{2/3}), has been shown to be closely related to the quality of cut and flow vertex sparsifiers. We study a variant of the 0-Extension problem where Steiner vertices are allowed. Specifically, we focus on the integrality gap of the same semi-metric LP relaxation to this new problem. Following from previous work, this new integrality gap turns out to be closely related to the quality achievable by cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph compression. We show that the new integrality gap stays superconstant Ω(log log |T|) even if we allow a super-linear O(|T|log^{1-ε}|T|) number of Steiner nodes.
Cite as

Yu Chen and Zihan Tan. Lower Bounds on 0-Extension with Steiner Nodes. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2024.47,
  author =	{Chen, Yu and Tan, Zihan},
  title =	{{Lower Bounds on 0-Extension with Steiner Nodes}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{47:1--47:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.47},
  URN =		{urn:nbn:de:0030-drops-201905},
  doi =		{10.4230/LIPIcs.ICALP.2024.47},
  annote =	{Keywords: Graph Algorithms, Zero Extension, Integrality Gap}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.50
Fully-Scalable MPC Algorithms for Clustering in High Dimension

Authors: Artur Czumaj, Guichen Gao, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We design new parallel algorithms for clustering in high-dimensional Euclidean spaces. These algorithms run in the Massively Parallel Computation (MPC) model, and are fully scalable, meaning that the local memory in each machine may be n^σ for arbitrarily small fixed σ > 0. Importantly, the local memory may be substantially smaller than the number of clusters k, yet all our algorithms are fast, i.e., run in O(1) rounds. We first devise a fast MPC algorithm for O(1)-approximation of uniform Facility Location. This is the first fully-scalable MPC algorithm that achieves O(1)-approximation for any clustering problem in general geometric setting; previous algorithms only provide poly(log n)-approximation or apply to restricted inputs, like low dimension or small number of clusters k; e.g. [Bhaskara and Wijewardena, ICML'18; Cohen-Addad et al., NeurIPS'21; Cohen-Addad et al., ICML'22]. We then build on this Facility Location result and devise a fast MPC algorithm that achieves O(1)-bicriteria approximation for k-Median and for k-Means, namely, it computes (1+ε)k clusters of cost within O(1/ε²)-factor of the optimum for k clusters. A primary technical tool that we introduce, and may be of independent interest, is a new MPC primitive for geometric aggregation, namely, computing for every data point a statistic of its approximate neighborhood, for statistics like range counting and nearest-neighbor search. Our implementation of this primitive works in high dimension, and is based on consistent hashing (aka sparse partition), a technique that was recently used for streaming algorithms [Czumaj et al., FOCS'22].
Cite as

Artur Czumaj, Guichen Gao, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý. Fully-Scalable MPC Algorithms for Clustering in High Dimension. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{czumaj_et_al:LIPIcs.ICALP.2024.50,
  author =	{Czumaj, Artur and Gao, Guichen and Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Vesel\'{y}, Pavel},
  title =	{{Fully-Scalable MPC Algorithms for Clustering in High Dimension}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.50},
  URN =		{urn:nbn:de:0030-drops-201938},
  doi =		{10.4230/LIPIcs.ICALP.2024.50},
  annote =	{Keywords: Massively parallel computing, high dimension, facility location, k-median, k-means}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.52
Simultaneously Approximating All 𝓁_p-Norms in Correlation Clustering

Authors: Sami Davies, Benjamin Moseley, and Heather Newman

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously O(1)-approximate for all 𝓁_p-norms of the disagreement vector; in other words, a combinatorial O(1)-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the 𝓁₂-norm objective, and more generally the first combinatorial algorithm for the 𝓁_p-norm objective when 1 < p < ∞. It is also faster than all previous algorithms that minimize the 𝓁_p-norm of the disagreement vector, with run-time O(n^ω), where O(n^ω) is the time for matrix multiplication on n × n matrices. When the maximum positive degree in the graph is at most Δ, this can be improved to a run-time of O(nΔ² log n).
Cite as

Sami Davies, Benjamin Moseley, and Heather Newman. Simultaneously Approximating All 𝓁_p-Norms in Correlation Clustering. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 52:1-52:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{davies_et_al:LIPIcs.ICALP.2024.52,
  author =	{Davies, Sami and Moseley, Benjamin and Newman, Heather},
  title =	{{Simultaneously Approximating All 𝓁\underlinep-Norms in Correlation Clustering}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{52:1--52:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.52},
  URN =		{urn:nbn:de:0030-drops-201950},
  doi =		{10.4230/LIPIcs.ICALP.2024.52},
  annote =	{Keywords: Approximation algorithms, correlation clustering, all-norms, lp-norms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.22
Two-Sets Cut-Uncut on Planar Graphs

Authors: Matthias Bentert, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Tuukka Korhonen

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We study Two-Sets Cut-Uncut on planar graphs. Therein, one is given an undirected planar graph G and two disjoint sets S and T of vertices as input. The question is, what is the minimum number of edges to remove from G, such that all vertices in S are separated from all vertices in T, while maintaining that every vertex in S, and respectively in T, stays in the same connected component. We show that this problem can be solved in 2^{|S|+|T|} n^𝒪(1) time with a one-sided-error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut is fixed-parameter tractable when parameterized by the number r of faces in a planar embedding covering the terminals S ∪ T, by providing a 2^𝒪(r) n^𝒪(1)-time algorithm.
Cite as

Matthias Bentert, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Tuukka Korhonen. Two-Sets Cut-Uncut on Planar Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bentert_et_al:LIPIcs.ICALP.2024.22,
  author =	{Bentert, Matthias and Drange, P\r{a}l Gr{\o}n\r{a}s and Fomin, Fedor V. and Golovach, Petr A. and Korhonen, Tuukka},
  title =	{{Two-Sets Cut-Uncut on Planar Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.22},
  URN =		{urn:nbn:de:0030-drops-201654},
  doi =		{10.4230/LIPIcs.ICALP.2024.22},
  annote =	{Keywords: planar graphs, cut-uncut, group-constrained paths}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.93
Streaming Algorithms for Connectivity Augmentation

Authors: Ce Jin, Michael Kapralov, Sepideh Mahabadi, and Ali Vakilian

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We study the k-connectivity augmentation problem (k-CAP) in the single-pass streaming model. Given a (k-1)-edge connected graph G = (V,E) that is stored in memory, and a stream of weighted edges (also called links) L with weights in {0,1,… ,W}, the goal is to choose a minimum weight subset L' ⊆ L of the links such that G' = (V,E∪ L') is k-edge connected. We give a (2+ε)-approximation algorithm for this problem which requires to store O(ε^{-1} nlog n) words. Moreover, we show the tightness of our result: Any algorithm with better than 2-approximation for the problem requires Ω(n²) bits of space even when k = 2. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for k-CAP. We further consider a natural generalization to the fully streaming model where both E and L arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a (2t-1+ε)-approximate weighted spanner of size at most O(ε^{-1} n^{1+1/t}log n) for integer t, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on log W. We believe that this result is of independent interest. Using our spanner result, we provide an optimal O(t)-approximation for k-CAP in the fully streaming model with O(nk + n^{1+1/t}) words of space. Finally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), k-edge connected spanning subgraph (k-ECSS) and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass O(tlog k)-approximation for SNDP using O(kn^{1+1/t}) words of space, where k is the maximum connectivity requirement.
Cite as

Ce Jin, Michael Kapralov, Sepideh Mahabadi, and Ali Vakilian. Streaming Algorithms for Connectivity Augmentation. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 93:1-93:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jin_et_al:LIPIcs.ICALP.2024.93,
  author =	{Jin, Ce and Kapralov, Michael and Mahabadi, Sepideh and Vakilian, Ali},
  title =	{{Streaming Algorithms for Connectivity Augmentation}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{93:1--93:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.93},
  URN =		{urn:nbn:de:0030-drops-202367},
  doi =		{10.4230/LIPIcs.ICALP.2024.93},
  annote =	{Keywords: streaming algorithms, connectivity augmentation}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.56
Light, Reliable Spanners

Authors: Arnold Filtser, Yuval Gitlitz, and Ofer Neiman

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

Abstract
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).
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.33
Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings

Authors: Arnold Filtser

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A (τ,ρ)-LSO is a collection Σ of orderings such that for every x,y ∈ ℝ^d there is an ordering σ ∈ Σ, where all the points between x and y w.r.t. σ are in the ρ-neighborhood of either x or y. In essence, LSO allow one to reduce problems to the 1-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO’s for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO’s for Euclidean, 𝓁_p, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO’s (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO’s to construct efficient labeled NNS data structures in this model.
Cite as

Arnold Filtser. Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser:LIPIcs.SoCG.2023.33,
  author =	{Filtser, Arnold},
  title =	{{Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{33:1--33:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.33},
  URN =		{urn:nbn:de:0030-drops-178839},
  doi =		{10.4230/LIPIcs.SoCG.2023.33},
  annote =	{Keywords: Locality sensitive ordering, nearest neighbor search, high dimensional Euclidean space, doubling dimension, planar and minor free graphs, path reporting low hop spanner, fault tolerant spanner, reliable spanner, light spanner}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.4
Communication Complexity of Inner Product in Symmetric Normed Spaces

Authors: Alexandr Andoni, Jarosław Błasiok, and Arnold Filtser

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm N on the space ℝⁿ. Here, Alice and Bob hold two vectors v,u such that ‖v‖_N ≤ 1 and ‖u‖_{N^*} ≤ 1, where N^* is the dual norm. The goal is to compute their inner product ⟨v,u⟩ up to an ε additive term. The problem is denoted by IP_N, and generalizes important previously studied problems, such as: (1) Computing the expectation 𝔼_{x∼𝒟}[f(x)] when Alice holds 𝒟 and Bob holds f is equivalent to IP_{𝓁₁}. (2) Computing v^TAv where Alice has a symmetric matrix with bounded operator norm (denoted S_∞) and Bob has a vector v where ‖v‖₂ = 1. This problem is complete for quantum communication complexity and is equivalent to IP_{S_∞}. We systematically study IP_N, showing the following results, near tight in most cases: 1) For any symmetric norm N, given ‖v‖_N ≤ 1 and ‖u‖_{N^*} ≤ 1 there is a randomized protocol using 𝒪̃(ε^{-6} log n) bits of communication that returns a value in ⟨u,v⟩±ε with probability 2/3 - we will denote this by ℛ_{ε,1/3}(IP_N) ≤ 𝒪̃(ε^{-6} log n). In a special case where N = 𝓁_p and N^* = 𝓁_q for p^{-1} + q^{-1} = 1, we obtain an improved bound ℛ_{ε,1/3}(IP_{𝓁_p}) ≤ 𝒪(ε^{-2} log n), nearly matching the lower bound ℛ_{ε, 1/3}(IP_{𝓁_p}) ≥ Ω(min(n, ε^{-2})). 2) One way communication complexity ℛ^{→}_{ε,δ}(IP_{𝓁_p}) ≤ 𝒪(ε^{-max(2,p)}⋅ log n/ε), and a nearly matching lower bound ℛ^{→}_{ε, 1/3}(IP_{𝓁_p}) ≥ Ω(ε^{-max(2,p)}) for ε^{-max(2,p)} ≪ n. 3) One way communication complexity ℛ^{→}_{ε,δ}(N) for a symmetric norm N is governed by the distortion of the embedding 𝓁_∞^k into N. Specifically, while a small distortion embedding easily implies a lower bound Ω(k), we show that, conversely, non-existence of such an embedding implies protocol with communication k^𝒪(log log k) log² n. 4) For arbitrary origin symmetric convex polytope P, we show ℛ_{ε,1/3}(IP_{N}) ≤ 𝒪(ε^{-2} log xc(P)), where N is the unique norm for which P is a unit ball, and xc(P) is the extension complexity of P (i.e. the smallest number of inequalities describing some polytope P' s.t. P is projection of P').
Cite as

Alexandr Andoni, Jarosław Błasiok, and Arnold Filtser. Communication Complexity of Inner Product in Symmetric Normed Spaces. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{andoni_et_al:LIPIcs.ITCS.2023.4,
  author =	{Andoni, Alexandr and B{\l}asiok, Jaros{\l}aw and Filtser, Arnold},
  title =	{{Communication Complexity of Inner Product in Symmetric Normed Spaces}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{4:1--4:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.4},
  URN =		{urn:nbn:de:0030-drops-175077},
  doi =		{10.4230/LIPIcs.ITCS.2023.4},
  annote =	{Keywords: communication complexity, symmetric norms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.50
Expander Decomposition in Dynamic Streams

Authors: Arnold Filtser, Michael Kapralov, and Mikhail Makarov

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning 𝒞 of vertices V such that the subgraphs of G induced by the clusters C ∈ 𝒞 are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (δ, ε)-multiplicative/additive error with high probability. The power cut sparsifier uses Õ(n/εδ) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant δ.
Cite as

Arnold Filtser, Michael Kapralov, and Mikhail Makarov. Expander Decomposition in Dynamic Streams. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.50,
  author =	{Filtser, Arnold and Kapralov, Michael and Makarov, Mikhail},
  title =	{{Expander Decomposition in Dynamic Streams}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{50:1--50:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.50},
  URN =		{urn:nbn:de:0030-drops-175534},
  doi =		{10.4230/LIPIcs.ITCS.2023.50},
  annote =	{Keywords: Streaming, expander decomposition, graph sparsifiers}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.18
Online Spanners in Metric Spaces

Authors: Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth

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

Abstract
Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.
Cite as

Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth. Online Spanners in Metric Spaces. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhore_et_al:LIPIcs.ESA.2022.18,
  author =	{Bhore, Sujoy and Filtser, Arnold and Khodabandeh, Hadi and T\'{o}th, Csaba D.},
  title =	{{Online Spanners in Metric Spaces}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{18:1--18:20},
  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.2022.18},
  URN =		{urn:nbn:de:0030-drops-169564},
  doi =		{10.4230/LIPIcs.ESA.2022.18},
  annote =	{Keywords: spanner, online algorithm, lightness, sparsity, minimum weight}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.47
Scattering and Sparse Partitions, and Their Applications

Authors: Arnold Filtser

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
A partition 𝒫 of a weighted graph G is (σ,τ,Δ)-sparse if every cluster has diameter at most Δ, and every ball of radius Δ/σ intersects at most τ clusters. Similarly, 𝒫 is (σ,τ,Δ)-scattering if instead for balls we require that every shortest path of length at most Δ/σ intersects at most τ clusters. Given a graph G that admits a (σ,τ,Δ)-sparse partition for all Δ > 0, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch O(τσ²log_τ n). Given a graph G that admits a (σ,τ,Δ)-scattering partition for all Δ > 0, we construct a solution for the Steiner Point Removal problem with stretch O(τ³σ³). We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.
Cite as

Arnold Filtser. Scattering and Sparse Partitions, and Their Applications. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filtser:LIPIcs.ICALP.2020.47,
  author =	{Filtser, Arnold},
  title =	{{Scattering and Sparse Partitions, and Their Applications}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{47:1--47:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.47},
  URN =		{urn:nbn:de:0030-drops-124547},
  doi =		{10.4230/LIPIcs.ICALP.2020.47},
  annote =	{Keywords: Scattering partitions, sparse partitions, sparse covers, Steiner point removal, Universal Steiner tree, Universal TSP}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.48
Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic

Authors: Arnold Filtser, Omrit Filtser, and Matthew J. Katz

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
In the (1+ε,r)-approximate near-neighbor problem for curves (ANNC) under some similarity measure δ, the goal is to construct a data structure for a given set 𝒞 of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ 𝒞 such that δ(Q,C)≤ r, then return a curve C' ∈ 𝒞 with δ(Q,C') ≤ (1+ε)r. There exists an efficient reduction from the (1+ε)-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C ∈ 𝒞 with δ(Q,C) ≤ (1+ε)⋅δ(Q,C^*), where C^* is the curve of 𝒞 most similar to Q. Given a set 𝒞 of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n⋅ O(1/ε)^{md} storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds. Further, we also consider the asymmetric version of ANNC, where the length of the query curves is k ≪ m, and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.
Cite as

Arnold Filtser, Omrit Filtser, and Matthew J. Katz. Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filtser_et_al:LIPIcs.ICALP.2020.48,
  author =	{Filtser, Arnold and Filtser, Omrit and Katz, Matthew J.},
  title =	{{Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{48:1--48:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.48},
  URN =		{urn:nbn:de:0030-drops-124555},
  doi =		{10.4230/LIPIcs.ICALP.2020.48},
  annote =	{Keywords: polygonal curves, Fr\'{e}chet distance, dynamic time warping, approximation algorithms, (asymmetric) approximate nearest neighbor, range counting}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2019.7
Towards a Theory of Parameterized Streaming Algorithms

Authors: Rajesh Chitnis and Graham Cormode

Published in: LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Abstract
Parameterized complexity attempts to give a more fine-grained analysis of the complexity of problems: instead of measuring the running time as a function of only the input size, we analyze the running time with respect to additional parameters. This approach has proven to be highly successful in delineating our understanding of NP-hard problems. Given this success with the TIME resource, it seems but natural to use this approach for dealing with the SPACE resource. First attempts in this direction have considered a few individual problems, with some success: Fafianie and Kratsch [MFCS'14] and Chitnis et al. [SODA'15] introduced the notions of streaming kernels and parameterized streaming algorithms respectively. For example, the latter shows how to refine the Omega(n^2) bit lower bound for finding a minimum Vertex Cover (VC) in the streaming setting by designing an algorithm for the parameterized k-VC problem which uses O(k^{2}log n) bits. In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms. We first define a natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS, and then obtain tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy (see Figure 1 and Table 1). On the algorithmic side, our parameterized streaming algorithms use techniques from the FPT world such as bidimensionality, iterative compression and bounded-depth search trees. On the hardness side, we obtain lower bounds for the parameterized streaming complexity of various problems via novel reductions from problems in communication complexity. We also show a general (unconditional) lower bound for space complexity of parameterized streaming algorithms for a large class of problems inspired by the recently developed frameworks for showing (conditional) kernelization lower bounds. Parameterized algorithms and streaming algorithms are approaches to cope with TIME and SPACE intractability respectively. It is our hope that this work on parameterized streaming algorithms leads to two-way flow of ideas between these two previously separated areas of theoretical computer science.
Cite as

Rajesh Chitnis and Graham Cormode. Towards a Theory of Parameterized Streaming Algorithms. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chitnis_et_al:LIPIcs.IPEC.2019.7,
  author =	{Chitnis, Rajesh and Cormode, Graham},
  title =	{{Towards a Theory of Parameterized Streaming Algorithms}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Jansen, Bart M. P. and Telle, Jan Arne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.7},
  URN =		{urn:nbn:de:0030-drops-114682},
  doi =		{10.4230/LIPIcs.IPEC.2019.7},
  annote =	{Keywords: Parameterized Algorithms, Streaming Algorithms, Kernels}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.6
On Strong Diameter Padded Decompositions

Authors: Arnold Filtser

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.
Cite as

Arnold Filtser. On Strong Diameter Padded Decompositions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{filtser:LIPIcs.APPROX-RANDOM.2019.6,
  author =	{Filtser, Arnold},
  title =	{{On Strong Diameter Padded Decompositions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{6:1--6:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.6},
  URN =		{urn:nbn:de:0030-drops-112217},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.6},
  annote =	{Keywords: Padded decomposition, Strong Diameter, Sparse Cover, Doubling Dimension, Minor free graphs, Unique Games, Spanners, Distance Oracles}
}
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