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

Let P be a set of n points in ℝ². For a parameter ε ∈ (0,1), a subset C ⊆ P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (1-ε)-factor in every direction. The set C is a weak ε-kernel of P if its directional width approximates that of P in every direction. Let 𝗄_ε(P) (resp. 𝗄^𝗐_ε(P)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an O(n 𝗄_ε(P)log n)-time algorithm for computing an ε-kernel of P of size 𝗄_ε(P), and an O(n²log n)-time algorithm for computing a weak ε-kernel of P of size 𝗄^𝗐_ε(P). We also present a fast algorithm for the Hausdorff variant of this problem.
In addition, we introduce the notion of ε-core, a convex polygon lying inside ch(P), prove that it is a good approximation of the optimal ε-kernel, present an efficient algorithm for computing it, and use it to compute an ε-kernel of small size.

Pankaj K. Agarwal and Sariel Har-Peled. Computing Instance-Optimal Kernels in Two Dimensions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 4:1-4:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.4, author = {Agarwal, Pankaj K. and Har-Peled, Sariel}, title = {{Computing Instance-Optimal Kernels in Two Dimensions}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {4:1--4:15}, 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.2023.4}, URN = {urn:nbn:de:0030-drops-178544}, doi = {10.4230/LIPIcs.SoCG.2023.4}, annote = {Keywords: Coreset, approximation, kernel} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We present a (1-ε)-approximation algorithms for maximum cardinality matchings in disk intersection graphs - all with near linear running time. We also present an estimation algorithm that returns (1±ε)-approximation to the size of such matchings - this algorithm runs in linear time for unit disks, and O(n log n) for general disks (as long as the density is relatively small).

Sariel Har-Peled and Everett Yang. Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 47:1-47:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2022.47, author = {Har-Peled, Sariel and Yang, Everett}, title = {{Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {47:1--47:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.47}, URN = {urn:nbn:de:0030-drops-160555}, doi = {10.4230/LIPIcs.SoCG.2022.47}, annote = {Keywords: Matchings, disk intersection graphs, approximation algorithms} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Tverberg’s theorem states that a set of n points in ℝ^d can be partitioned into ⌈n/(d+1)⌉ sets whose convex hulls all intersect. A point in the intersection (aka Tverberg point) is a centerpoint, or high-dimensional median, of the input point set. While randomized algorithms exist to find centerpoints with some failure probability, a partition for a Tverberg point provides a certificate of its correctness.
Unfortunately, known algorithms for computing exact Tverberg points take n^{O(d²)} time. We provide several new approximation algorithms for this problem, which improve running time or approximation quality over previous work. In particular, we provide the first strongly polynomial (in both n and d) approximation algorithm for finding a Tverberg point.

Sariel Har-Peled and Timothy Zhou. Improved Approximation Algorithms for Tverberg Partitions. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 51:1-51:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{harpeled_et_al:LIPIcs.ESA.2021.51, author = {Har-Peled, Sariel and Zhou, Timothy}, title = {{Improved Approximation Algorithms for Tverberg Partitions}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.51}, URN = {urn:nbn:de:0030-drops-146323}, doi = {10.4230/LIPIcs.ESA.2021.51}, annote = {Keywords: Geometric spanners, vertex failures, robustness} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We study colored coverage and clustering problems. Here, we are given a colored point set, where the points are covered by k (unknown) clusters, which are monochromatic (i.e., all the points covered by the same cluster have the same color). The access to the colors of the points (or even the points themselves) is provided indirectly via various oracle queries (such as nearest neighbor, or separation queries). We show that one can correctly deduce the color of all the points (i.e., compute a monochromatic clustering of the points) using a polylogarithmic number of queries, if the number of clusters is a constant.
We investigate several variants of this problem, including Undecided Linear Programming and covering of points by k monochromatic balls.

Stav Ashur and Sariel Har-Peled. On Undecided LP, Clustering and Active Learning. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 12:1-12:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ashur_et_al:LIPIcs.SoCG.2021.12, author = {Ashur, Stav and Har-Peled, Sariel}, title = {{On Undecided LP, Clustering and Active Learning}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.12}, URN = {urn:nbn:de:0030-drops-138116}, doi = {10.4230/LIPIcs.SoCG.2021.12}, annote = {Keywords: Linear Programming, Active learning, Classification} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We study the problem of constructing weak ε-nets where the stabbing elements are lines or k-flats instead of points. We study this problem in the simplest setting where it is still interesting - namely, the uniform measure of volume over the hypercube [0,1]^d. Specifically, a (k,ε)-net is a set of k-flats, such that any convex body in [0,1]^d of volume larger than ε is stabbed by one of these k-flats. We show that for k ≥ 1, one can construct (k,ε)-nets of size O(1/ε^{1-k/d}). We also prove that any such net must have size at least Ω(1/ε^{1-k/d}). As a concrete example, in three dimensions all ε-heavy bodies in [0,1]³ can be stabbed by Θ(1/ε^{2/3}) lines. Note, that these bounds are sublinear in 1/ε, and are thus somewhat surprising.

Sariel Har-Peled and Mitchell Jones. Stabbing Convex Bodies with Lines and Flats. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 42:1-42:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2021.42, author = {Har-Peled, Sariel and Jones, Mitchell}, title = {{Stabbing Convex Bodies with Lines and Flats}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {42:1--42:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.42}, URN = {urn:nbn:de:0030-drops-138412}, doi = {10.4230/LIPIcs.SoCG.2021.42}, annote = {Keywords: Discrete geometry, combinatorics, weak \epsilon-nets, k-flats} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation, that is, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.

Sariel Har-Peled, Manor Mendel, and Dániel Oláh. Reliable Spanners for Metric Spaces. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 43:1-43:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2021.43, author = {Har-Peled, Sariel and Mendel, Manor and Ol\'{a}h, D\'{a}niel}, title = {{Reliable Spanners for Metric Spaces}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {43:1--43:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.43}, URN = {urn:nbn:de:0030-drops-138423}, doi = {10.4230/LIPIcs.SoCG.2021.43}, annote = {Keywords: Spanners, reliability} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Reliable spanners can withstand huge failures, even when a linear number of vertices are deleted from the network. In case of failures, some of the remaining vertices of a reliable spanner may no longer admit the spanner property, but this collateral damage is bounded by a fraction of the size of the attack. It is known that Ω(nlog n) edges are needed to achieve this strong property, where n is the number of vertices in the network, even in one dimension. Constructions of reliable geometric (1+ε)-spanners, for n points in ℝ^d, are known, where the resulting graph has 𝒪(n log n log log⁶n) edges.
Here, we show randomized constructions of smaller size spanners that have the desired reliability property in expectation or with good probability. The new construction is simple, and potentially practical - replacing a hierarchical usage of expanders (which renders the previous constructions impractical) by a simple skip list like construction. This results in a 1-spanner, on the line, that has linear number of edges. Using this, we present a construction of a reliable spanner in ℝ^d with 𝒪(n log log²n log log log n) edges.

Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. Sometimes Reliable Spanners of Almost Linear Size. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 27:1-27:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2020.27, author = {Buchin, Kevin and Har-Peled, Sariel and Ol\'{a}h, D\'{a}niel}, title = {{Sometimes Reliable Spanners of Almost Linear Size}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.27}, URN = {urn:nbn:de:0030-drops-128934}, doi = {10.4230/LIPIcs.ESA.2020.27}, annote = {Keywords: Geometric spanners, vertex failures, reliability} }

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Track A: Algorithms, Complexity and Games

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

Consider a set P ⊆ ℝ^d of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O(⬡_P log n) queries, where ⬡_P is the largest subset of points of P in convex position. In 2D, we provide an algorithm which efficiently generates these adaptive queries.
Furthermore, we show that in two dimensions one can solve this problem using O(⊚(P,C) log² n) oracle queries, where ⊚(P,C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires. Finally, we consider other variations on the problem, such as using the fewest number of queries to decide if C contains all points of P.
As an application of the above, we show that the discrete geometric median of a point set P in ℝ² can be computed in O(n log² n (log n log log n + ⬡(P))) expected time.

Sariel Har-Peled, Mitchell Jones, and Saladi Rahul. Active Learning a Convex Body in Low Dimensions. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 64:1-64:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{harpeled_et_al:LIPIcs.ICALP.2020.64, author = {Har-Peled, Sariel and Jones, Mitchell and Rahul, Saladi}, title = {{Active Learning a Convex Body in Low Dimensions}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {64:1--64:17}, 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.2020.64}, URN = {urn:nbn:de:0030-drops-124711}, doi = {10.4230/LIPIcs.ICALP.2020.64}, annote = {Keywords: Approximation algorithms, computational geometry, separation oracles, active learning} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We study a clustering problem where the goal is to maximize the coverage of the input points by k chosen centers. Specifically, given a set of n points P ⊆ ℝ^d, the goal is to pick k centers C ⊆ ℝ^d that maximize the service ∑_{p∈P}φ(𝖽(p,C)) to the points P, where 𝖽(p,C) is the distance of p to its nearest center in C, and φ is a non-increasing service function φ: ℝ+ → ℝ+. This includes problems of placing k base stations as to maximize the total bandwidth to the clients - indeed, the closer the client is to its nearest base station, the more data it can send/receive, and the target is to place k base stations so that the total bandwidth is maximized. We provide an n^{ε^-O(d)} time algorithm for this problem that achieves a (1-ε)-approximation. Notably, the runtime does not depend on the parameter k and it works for an arbitrary non-increasing service function φ: ℝ+ → ℝ+.

Arturs Backurs and Sariel Har-Peled. Submodular Clustering in Low Dimensions. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{backurs_et_al:LIPIcs.SWAT.2020.8, author = {Backurs, Arturs and Har-Peled, Sariel}, title = {{Submodular Clustering in Low Dimensions}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.8}, URN = {urn:nbn:de:0030-drops-122551}, doi = {10.4230/LIPIcs.SWAT.2020.8}, annote = {Keywords: clustering, covering, PTAS} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Let P be a set of n points in ℝ^d in general position. A median hyperplane (roughly) splits the point set P in half. The yolk of P is the ball of smallest radius intersecting all median hyperplanes of P. The egg of P is the ball of smallest radius intersecting all hyperplanes which contain exactly d points of P.
We present exact algorithms for computing the yolk and the egg of a point set, both running in expected time O(n^(d-1) log n). The running time of the new algorithm is a polynomial time improvement over existing algorithms. We also present algorithms for several related problems, such as computing the Tukey and center balls of a point set, among others.

Sariel Har-Peled and Mitchell Jones. Fast Algorithms for Geometric Consensuses. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 50:1-50:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2020.50, author = {Har-Peled, Sariel and Jones, Mitchell}, title = {{Fast Algorithms for Geometric Consensuses}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {50:1--50:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.50}, URN = {urn:nbn:de:0030-drops-122088}, doi = {10.4230/LIPIcs.SoCG.2020.50}, annote = {Keywords: Geometric optimization, centerpoint, voting games} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We show how to construct (1+epsilon)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters theta, epsilon in (0,1), the computed spanner G has O(epsilon^{-7d} log^7 epsilon^{-1} * theta^{-6} n log n (log log n)^6) edges. Furthermore, for any k, and any deleted set B subseteq P of k points, the residual graph G \ B is (1+epsilon)-spanner for all the points of P except for (1+theta)k of them. No previous constructions, beyond the trivial clique with O(n^2) edges, were known such that only a tiny additional fraction (i.e., theta) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one dimensional construction in a black box fashion.

Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. A Spanner for the Day After. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 19:1-19:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2019.19, author = {Buchin, Kevin and Har-Peled, Sariel and Ol\'{a}h, D\'{a}niel}, title = {{A Spanner for the Day After}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.19}, URN = {urn:nbn:de:0030-drops-104237}, doi = {10.4230/LIPIcs.SoCG.2019.19}, annote = {Keywords: Geometric spanners, vertex failures, robustness} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Given a set of n points in the plane, and a parameter k, we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing k points. We present the first near quadratic time algorithm for this problem, improving over the previous near-O(n^{5/2})-time algorithm by Kaplan et al. [Haim Kaplan et al., 2017]. We provide an almost matching conditional lower bound, under the assumption that (min,+)-convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for either perimeter or area) that can make the time bound sensitive to k, giving near O(n k) time. We also present a near linear time (1+epsilon)-approximation algorithm to the minimum area of the optimal rectangle containing k points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

Timothy M. Chan and Sariel Har-Peled. Smallest k-Enclosing Rectangle Revisited. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2019.23, author = {Chan, Timothy M. and Har-Peled, Sariel}, title = {{Smallest k-Enclosing Rectangle Revisited}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.23}, URN = {urn:nbn:de:0030-drops-104276}, doi = {10.4230/LIPIcs.SoCG.2019.23}, annote = {Keywords: Geometric optimization, outliers, approximation algorithms, conditional lower bounds} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We revisit an algorithm of Clarkson et al. [K. L. Clarkson et al., 1996], that computes (roughly) a 1/(4d^2)-centerpoint in O~(d^9) time, for a point set in R^d, where O~ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time O~(d^7). While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.

Sariel Har-Peled and Mitchell Jones. Journey to the Center of the Point Set. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 41:1-41:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2019.41, author = {Har-Peled, Sariel and Jones, Mitchell}, title = {{Journey to the Center of the Point Set}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.41}, URN = {urn:nbn:de:0030-drops-104454}, doi = {10.4230/LIPIcs.SoCG.2019.41}, annote = {Keywords: Computational geometry, Centerpoints, Random walks} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

For any constant d and parameter epsilon > 0, we show the existence of (roughly) 1/epsilon^d orderings on the unit cube [0,1)^d, such that any two points p, q in [0,1)^d that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between p and q in the ordering are points with Euclidean distance at most epsilon | p - q | from p or q. These orderings are extensions of the Z-order, and they can be efficiently computed.
Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.

Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On Locality-Sensitive Orderings and Their Applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 21:1-21:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chan_et_al:LIPIcs.ITCS.2019.21, author = {Chan, Timothy M. and Har-Peled, Sariel and Jones, Mitchell}, title = {{On Locality-Sensitive Orderings and Their Applications}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {21:1--21:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.21}, URN = {urn:nbn:de:0030-drops-101140}, doi = {10.4230/LIPIcs.ITCS.2019.21}, annote = {Keywords: Approximation algorithms, Data structures, Computational geometry} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points.
This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.

Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert. Stabbing Pairwise Intersecting Disks by Five Points. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 50:1-50:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{harpeled_et_al:LIPIcs.ISAAC.2018.50, author = {Har-Peled, Sariel and Kaplan, Haim and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha and Willert, Max}, title = {{Stabbing Pairwise Intersecting Disks by Five Points}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {50:1--50:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.50}, URN = {urn:nbn:de:0030-drops-99989}, doi = {10.4230/LIPIcs.ISAAC.2018.50}, annote = {Keywords: Disk graph, piercing set, LP-type problem} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

In the Sparse Linear Regression (SLR) problem, given a d x n matrix M and a d-dimensional query q, the goal is to compute a k-sparse n-dimensional vector tau such that the error ||M tau - q|| is minimized. This problem is equivalent to the following geometric problem: given a set P of n points and a query point q in d dimensions, find the closest k-dimensional subspace to q, that is spanned by a subset of k points in P. In this paper, we present data-structures/algorithms and conditional lower bounds for several variants of this problem (such as finding the closest induced k dimensional flat/simplex instead of a subspace).
In particular, we present approximation algorithms for the online variants of the above problems with query time O~(n^{k-1}), which are of interest in the "low sparsity regime" where k is small, e.g., 2 or 3. For k=d, this matches, up to polylogarithmic factors, the lower bound that relies on the affinely degenerate conjecture (i.e., deciding if n points in R^d contains d+1 points contained in a hyperplane takes Omega(n^d) time). Moreover, our algorithms involve formulating and solving several geometric subproblems, which we believe to be of independent interest.

Sariel Har-Peled, Piotr Indyk, and Sepideh Mahabadi. Approximate Sparse Linear Regression. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 77:1-77:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{harpeled_et_al:LIPIcs.ICALP.2018.77, author = {Har-Peled, Sariel and Indyk, Piotr and Mahabadi, Sepideh}, title = {{Approximate Sparse Linear Regression}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {77:1--77:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.77}, URN = {urn:nbn:de:0030-drops-90816}, doi = {10.4230/LIPIcs.ICALP.2018.77}, annote = {Keywords: Sparse Linear Regression, Approximate Nearest Neighbor, Sparse Recovery, Nearest Induced Flat, Nearest Subspace Search} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only polylog(n) bipartite independent set queries, and another one that uses n^{2/3} polylog(n) independent set queries.

Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge Estimation with Independent Set Oracles. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 38:1-38:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beame_et_al:LIPIcs.ITCS.2018.38, author = {Beame, Paul and Har-Peled, Sariel and Natarajan Ramamoorthy, Sivaramakrishnan and Rashtchian, Cyrus and Sinha, Makrand}, title = {{Edge Estimation with Independent Set Oracles}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {38:1--38:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.38}, URN = {urn:nbn:de:0030-drops-83552}, doi = {10.4230/LIPIcs.ITCS.2018.38}, annote = {Keywords: Approximate Counting, Independent Set Queries, Sparsification, Importance Sampling} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Given a set P of n points in R^d , we show how to insert a set Z of O(n^(1-1/d)) additional points, such that P can be broken into two sets P1 and P2 , of roughly equal size, such that in the Voronoi diagram V(P u Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1,P2) of P , we present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition.

Vijay V. S. P. Bhattiprolu and Sariel Har-Peled. Separating a Voronoi Diagram via Local Search. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 18:1-18:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bhattiprolu_et_al:LIPIcs.SoCG.2016.18, author = {Bhattiprolu, Vijay V. S. P. and Har-Peled, Sariel}, title = {{Separating a Voronoi Diagram via Local Search}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.18}, URN = {urn:nbn:de:0030-drops-59107}, doi = {10.4230/LIPIcs.SoCG.2016.18}, annote = {Keywords: Separators, Local search, Approximation, Voronoi diagrams, Delaunay triangulation, Meshing, Geometric hitting set} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon P with n vertices in a read only memory, and additional working memory of size m, the new algorithm computes the shortest path (in P) in O(n^2 / m) expected time, assuming m = O(n / log^2 n). This requires several new tools, which we believe to be of independent interest.
Specifically, we show that violator space problems, an abstraction of low dimensional linear-programming (and LP-type problems), can be solved using constant space and expected linear time, by modifying Seidel's linear programming algorithm and using pseudo-random sequences.

Sariel Har-Peled. Shortest Path in a Polygon using Sublinear Space. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 111-125, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{harpeled:LIPIcs.SOCG.2015.111, author = {Har-Peled, Sariel}, title = {{Shortest Path in a Polygon using Sublinear Space}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {111--125}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.111}, URN = {urn:nbn:de:0030-drops-50941}, doi = {10.4230/LIPIcs.SOCG.2015.111}, annote = {Keywords: Shortest path, violator spaces, limited space} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We investigate what computational tasks can be performed on a point set in R^d, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following:
(A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set.
(B) One can decide if a query point is (approximately) inside the convex-hull of the point set.
We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.

Sariel Har-Peled, Nirman Kumar, David M. Mount, and Benjamin Raichel. Space Exploration via Proximity Search. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 374-389, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{harpeled_et_al:LIPIcs.SOCG.2015.374, author = {Har-Peled, Sariel and Kumar, Nirman and Mount, David M. and Raichel, Benjamin}, title = {{Space Exploration via Proximity Search}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {374--389}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.374}, URN = {urn:nbn:de:0030-drops-51004}, doi = {10.4230/LIPIcs.SOCG.2015.374}, annote = {Keywords: Proximity search, implicit point set, probing} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We present an extension of Voronoi diagrams where not only the distance to the site is taken into account when considering which site the client is going to use, but additional attributes (i.e., prices or weights) are also considered. A cell in this diagram is then the loci of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest.

Hsien-Chih Chang, Sariel Har-Peled, and Benjamin Raichel. From Proximity to Utility: A Voronoi Partition of Pareto Optima. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 689-703, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chang_et_al:LIPIcs.SOCG.2015.689, author = {Chang, Hsien-Chih and Har-Peled, Sariel and Raichel, Benjamin}, title = {{From Proximity to Utility: A Voronoi Partition of Pareto Optima}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {689--703}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.689}, URN = {urn:nbn:de:0030-drops-50925}, doi = {10.4230/LIPIcs.SOCG.2015.689}, annote = {Keywords: Voronoi diagrams, expected complexity, backward analysis, Pareto optima, candidate diagram, Clarkson-Shor technique} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Given a set of n disjoint balls b_1, ..., b_n in R^d, we provide a data structure, of near linear size, that can answer (1 +- epsilon)-approximate k-th-nearest neighbor queries in O(log(n) + 1/epsilon^d) time, where k and epsilon are provided at query time. If k and epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large.

Sariel Har-Peled and Nirman Kumar. Robust Proximity Search for Balls Using Sublinear Space. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 315-326, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{harpeled_et_al:LIPIcs.FSTTCS.2014.315, author = {Har-Peled, Sariel and Kumar, Nirman}, title = {{Robust Proximity Search for Balls Using Sublinear Space}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {315--326}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.315}, URN = {urn:nbn:de:0030-drops-48526}, doi = {10.4230/LIPIcs.FSTTCS.2014.315}, annote = {Keywords: Approximate Nearest neighbors, algorithms, data structures} }

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