28 Search Results for "Har-Peled, Sariel"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.4
Computing Instance-Optimal Kernels in Two Dimensions

Authors: Pankaj K. Agarwal and Sariel Har-Peled

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

Abstract
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.
Cite as

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-dev.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}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2023.64
Greedy Permutations and Finite Voronoi Diagrams (Media Exposition)

Authors: Oliver A. Chubet, Paul Macnichol, Parth Parikh, Donald R. Sheehy, and Siddharth S. Sheth

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

Abstract
We illustrate the computation of a greedy permutation using finite Voronoi diagrams. We describe the neighbor graph, which is a sparse graph data structure that facilitates efficient point location to insert a new Voronoi cell. This data structure is not dependent on a Euclidean metric space. The greedy permutation is computed in O(nlog Δ) time for low-dimensional data using this method [Sariel Har-Peled and Manor Mendel, 2006; Donald R. Sheehy, 2020].
Cite as

Oliver A. Chubet, Paul Macnichol, Parth Parikh, Donald R. Sheehy, and Siddharth S. Sheth. Greedy Permutations and Finite Voronoi Diagrams (Media Exposition). In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 64:1-64:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chubet_et_al:LIPIcs.SoCG.2023.64,
  author =	{Chubet, Oliver A. and Macnichol, Paul and Parikh, Parth and Sheehy, Donald R. and Sheth, Siddharth S.},
  title =	{{Greedy Permutations and Finite Voronoi Diagrams}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{64:1--64:5},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.64},
  URN =		{urn:nbn:de:0030-drops-179146},
  doi =		{10.4230/LIPIcs.SoCG.2023.64},
  annote =	{Keywords: greedy permutation, Voronoi diagrams}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.47
Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs

Authors: Sariel Har-Peled and Everett Yang

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

Abstract
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).
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.26
Fast and Exact Convex Hull Simplification

Authors: Georgiy Klimenko and Benjamin Raichel

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
Given a point set P in the plane, we seek a subset Q ⊆ P, whose convex hull gives a smaller and thus simpler representation of the convex hull of P. Specifically, let cost(Q,P) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q,P) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q,P), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log²n) time algorithm and an O(n log³n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n^2.5302) time algorithm when minimizing k and an O(min{n^2.5302, kn^2.376}) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.
Cite as

Georgiy Klimenko and Benjamin Raichel. Fast and Exact Convex Hull Simplification. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{klimenko_et_al:LIPIcs.FSTTCS.2021.26,
  author =	{Klimenko, Georgiy and Raichel, Benjamin},
  title =	{{Fast and Exact Convex Hull Simplification}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.26},
  URN =		{urn:nbn:de:0030-drops-155373},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.26},
  annote =	{Keywords: Convex hull, coreset, exact algorithm}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.51
Improved Approximation Algorithms for Tverberg Partitions

Authors: Sariel Har-Peled and Timothy Zhou

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.12
On Undecided LP, Clustering and Active Learning

Authors: Stav Ashur and Sariel Har-Peled

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

Abstract
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.
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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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.42
Stabbing Convex Bodies with Lines and Flats

Authors: Sariel Har-Peled and Mitchell Jones

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

Abstract
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.
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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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.43
Reliable Spanners for Metric Spaces

Authors: Sariel Har-Peled, Manor Mendel, and Dániel Oláh

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.27
Sometimes Reliable Spanners of Almost Linear Size

Authors: Kevin Buchin, Sariel Har-Peled, and Dániel Oláh

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.45
Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams

Authors: Chenglin Fan and Benjamin Raichel

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

Abstract
While the standard unweighted Voronoi diagram in the plane has linear worst-case complexity, many of its natural generalizations do not. This paper considers two such previously studied generalizations, namely multiplicative and semi Voronoi diagrams. These diagrams both have quadratic worst-case complexity, though here we show that their expected complexity is linear for certain natural randomized inputs. Specifically, we argue that the expected complexity is linear for: (1) semi Voronoi diagrams when the visible direction is randomly sampled, and (2) for multiplicative diagrams when either weights are sampled from a constant-sized set, or the more challenging case when weights are arbitrary but locations are sampled from a square.
Cite as

Chenglin Fan and Benjamin Raichel. Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fan_et_al:LIPIcs.ESA.2020.45,
  author =	{Fan, Chenglin and Raichel, Benjamin},
  title =	{{Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{45:1--45:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.45},
  URN =		{urn:nbn:de:0030-drops-129111},
  doi =		{10.4230/LIPIcs.ESA.2020.45},
  annote =	{Keywords: Voronoi Diagrams, Expected Complexity, Computational Geometry}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.64
Active Learning a Convex Body in Low Dimensions

Authors: Sariel Har-Peled, Mitchell Jones, and Saladi Rahul

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2020.8
Submodular Clustering in Low Dimensions

Authors: Arturs Backurs and Sariel Har-Peled

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

Abstract
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+.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.50
Fast Algorithms for Geometric Consensuses

Authors: Sariel Har-Peled and Mitchell Jones

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.19
A Spanner for the Day After

Authors: Kevin Buchin, Sariel Har-Peled, and Dániel Oláh

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.20
Computing Shapley Values in the Plane

Authors: Sergio Cabello and Timothy M. Chan

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

Abstract
We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box. For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods. We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points.
Cite as

Sergio Cabello and Timothy M. Chan. Computing Shapley Values in the Plane. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2019.20,
  author =	{Cabello, Sergio and Chan, Timothy M.},
  title =	{{Computing Shapley Values in the Plane}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{20:1--20:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.20},
  URN =		{urn:nbn:de:0030-drops-104244},
  doi =		{10.4230/LIPIcs.SoCG.2019.20},
  annote =	{Keywords: Shapley values, stochastic computational geometry, convex hull, minimum enclosing disk, bounding box, arrangements, convolutions, airport problem}
}
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