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Documents authored by Dutta, Kunal

Document
DOI: 10.4230/LIPIcs.ESA.2024.29
A Euclidean Embedding for Computing Persistent Homology with Gaussian Kernels

Authors: Jean-Daniel Boissonnat and Kunal Dutta

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
Computing persistent homology of large datasets using Gaussian kernels is useful in the domains of topological data analysis and machine learning as shown by Phillips, Wang and Zheng [SoCG 2015]. However, unlike in the case of persistent homology computation using the Euclidean distance or the k-distance, using Gaussian kernels involves significantly higher overhead, as all distance computations are in terms of the Gaussian kernel distance which is computationally more expensive. Further, most algorithmic implementations (e.g. Gudhi, Ripser, etc.) are based on Euclidean distances, so the question of finding a Euclidean embedding - preferably low-dimensional - that preserves the persistent homology computed with Gaussian kernels, is quite important. We consider the Gaussian kernel power distance (GKPD) given by Phillips, Wang and Zheng. Given an n-point dataset and a relative error parameter {ε} ∈ (0,1], we show that the persistent homology of the {Čech } filtration of the dataset computed using the GKPD can be approximately preserved using O({ε}^{-2}log n) dimensions, under a high stable rank condition. Our results also extend to the Delaunay filtration and the (simpler) case of the weighted Rips filtrations constructed using the GKPD. Compared to the Euclidean embedding for the Gaussian kernel function in ∼ n dimensions, which uses the Cholesky decomposition of the matrix of the kernel function applied to all pairs of data points, our embedding may also be viewed as dimensionality reduction - reducing the dimensionality from n to ∼ log n dimensions. Our proof utilizes the embedding of Chen and Phillips [ALT 2017], based on the Random Fourier Functions of Rahimi and Recht [NeurIPS 2007], together with two novel ingredients. The first one is a new decomposition of the squared radii of {Čech } simplices computed using the GKPD, in terms of the pairwise GKPDs between the vertices, which we state and prove. The second is a new concentration inequality for sums of cosine functions of Gaussian random vectors, which we call Gaussian cosine chaoses. We believe these are of independent interest and will find other applications in future.
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Jean-Daniel Boissonnat and Kunal Dutta. A Euclidean Embedding for Computing Persistent Homology with Gaussian Kernels. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boissonnat_et_al:LIPIcs.ESA.2024.29,
  author =	{Boissonnat, Jean-Daniel and Dutta, Kunal},
  title =	{{A Euclidean Embedding for Computing Persistent Homology with Gaussian Kernels}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.29},
  URN =		{urn:nbn:de:0030-drops-211009},
  doi =		{10.4230/LIPIcs.ESA.2024.29},
  annote =	{Keywords: Persistent homology, Gaussian kernels, Random Fourier Features, Euclidean embedding}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.21
On Edge Collapse of Random Simplicial Complexes

Authors: Jean-Daniel Boissonnat, Kunal Dutta, Soumik Dutta, and Siddharth Pritam

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

Abstract
We consider the edge collapse (introduced in [Boissonnat, Pritam. SoCG 2020]) process on the Erdős-Rényi random clique complex X(n,c/√n) on n vertices with edge probability c/√n such that c > √η₂ where η₂ = inf{η | x = e^{-η(1-x)²} has a solution in (0,1)}. For a given c > √η₂, we show that after t iterations of maximal edge collapsing phases, the remaining subcomplex, or t-core, has at most (1+o(1))binom(n,2)p(1-(c²/3)(1-(1-γ_t)³)) and at least (1+o(1)) binom(n,2) p(1-γ_{t+1}-c²γ_t(1-γ_t)²) edges asymptotically almost surely (a.a.s.), where {γ_t}_{t ≥ 0} is recursively determined by γ_{t+1} = e^{-c²(1-γ_t)²} and γ_0 = 0. We also determine the upper and lower bound on the final core with explicit formulas. If c < √{η₂} then we show that the final core contains o(n√n) edges. On the other hand, if, instead of c being a constant with respect to n, c > √{2log n} then the edge collapse process is no more effective in reducing the size of the complex. Our proof is based on the notion of local weak convergence [Aldous, Steele. In Probability on discrete structures. Springer, 2004] together with two new components. Firstly, we identify the critical combinatorial structures that control the outcome of the edge collapse process. By controlling the expected number of these structures during the edge collapse process we establish a.a.s. bounds on the size of the core. We also give a new concentration inequality for typically Lipschitz functions on random graphs which improves on the bound of [Warnke. Combinatorics, Probability and Computing, 2016] and is, therefore, of independent interest. The proof of our lower bound is via the recursive technique of [Linial and Peled. A Journey Through Discrete Mathematics. 2017] to simulate cycles in infinite trees. These are the first theoretical results proved for edge collapses on random (or non-random) simplicial complexes.
Cite as

Jean-Daniel Boissonnat, Kunal Dutta, Soumik Dutta, and Siddharth Pritam. On Edge Collapse of Random Simplicial Complexes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2024.21,
  author =	{Boissonnat, Jean-Daniel and Dutta, Kunal and Dutta, Soumik and Pritam, Siddharth},
  title =	{{On Edge Collapse of Random Simplicial Complexes}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{21:1--21:16},
  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.2024.21},
  URN =		{urn:nbn:de:0030-drops-199661},
  doi =		{10.4230/LIPIcs.SoCG.2024.21},
  annote =	{Keywords: Computational Topology, Topological Data Analysis, Strong Collapse, Simple Collapse, Persistent homology, Random simplicial complexes}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.59
Uniform Brackets, Containers, and Combinatorial Macbeath Regions

Authors: Kunal Dutta, Arijit Ghosh, and Shay Moran

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We study the connections between three seemingly different combinatorial structures - uniform brackets in statistics and probability theory, containers in online and distributed learning theory, and combinatorial Macbeath regions, or Mnets in discrete and computational geometry. We show that these three concepts are manifestations of a single combinatorial property that can be expressed under a unified framework along the lines of Vapnik-Chervonenkis type theory for uniform convergence. These new connections help us to bring tools from discrete and computational geometry to prove improved bounds for these objects. Our improved bounds help to get an optimal algorithm for distributed learning of halfspaces, an improved algorithm for the distributed convex set disjointness problem, and improved regret bounds for online algorithms against σ-smoothed adversary for a large class of semi-algebraic threshold functions.
Cite as

Kunal Dutta, Arijit Ghosh, and Shay Moran. Uniform Brackets, Containers, and Combinatorial Macbeath Regions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 59:1-59:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dutta_et_al:LIPIcs.ITCS.2022.59,
  author =	{Dutta, Kunal and Ghosh, Arijit and Moran, Shay},
  title =	{{Uniform Brackets, Containers, and Combinatorial Macbeath Regions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{59:1--59:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.59},
  URN =		{urn:nbn:de:0030-drops-156551},
  doi =		{10.4230/LIPIcs.ITCS.2022.59},
  annote =	{Keywords: communication complexity, distributed learning, emperical process theory, online algorithms, discrete geometry, computational geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.10
Dimensionality Reduction for k-Distance Applied to Persistent Homology

Authors: Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta, and Martin Lotz

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

Abstract
Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy [Proc. SoCG, 2014]. We show that any linear transformation that preserves pairwise distances up to a (1±ε) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of (1-ε)^{-1}. Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. are preserved up to a (1±ε) factor. We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional manifold, obtaining the target dimension bounds of Lotz [Proc. Roy. Soc. , 2019] and Clarkson [Proc. SoCG, 2008 ] respectively.
Cite as

Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta, and Martin Lotz. Dimensionality Reduction for k-Distance Applied to Persistent Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arya_et_al:LIPIcs.SoCG.2020.10,
  author =	{Arya, Shreya and Boissonnat, Jean-Daniel and Dutta, Kunal and Lotz, Martin},
  title =	{{Dimensionality Reduction for k-Distance Applied to Persistent Homology}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{10:1--10:15},
  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.2020.10},
  URN =		{urn:nbn:de:0030-drops-121682},
  doi =		{10.4230/LIPIcs.SoCG.2020.10},
  annote =	{Keywords: Dimensionality reduction, Johnson-Lindenstrauss lemma, Topological Data Analysis, Persistent Homology, k-distance, distance to measure}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.22
Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets

Authors: Jean-Daniel Boissonnat, Olivier Devillers, Kunal Dutta, and Marc Glisse

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice. Randomized incremental constructions are most of the time space and time optimal in the worst-case, as exemplified by the construction of convex hulls, Delaunay triangulations and arrangements of line segments. However, the worst-case scenario occurs rarely in practice and we would like to understand how RIC behaves when the input is nice in the sense that the associated output is significantly smaller than in the worst-case. For example, it is known that the Delaunay triangulations of nicely distributed points on polyhedral surfaces in E^3 has linear complexity, as opposed to a worst-case quadratic complexity. The standard analysis does not provide accurate bounds on the complexity of such cases and we aim at establishing such bounds in this paper. More precisely, we will show that, in the case of nicely distributed points on polyhedral surfaces, the complexity of the usual RIC is O(n log n), which is optimal. In other words, without any modification, RIC nicely adapts to good cases of practical value. Our proofs also work for some other notions of nicely distributed point sets, such as (epsilon, kappa)-samples. Along the way, we prove a probabilistic lemma for sampling without replacement, which may be of independent interest.
Cite as

Jean-Daniel Boissonnat, Olivier Devillers, Kunal Dutta, and Marc Glisse. Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{boissonnat_et_al:LIPIcs.ESA.2019.22,
  author =	{Boissonnat, Jean-Daniel and Devillers, Olivier and Dutta, Kunal and Glisse, Marc},
  title =	{{Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{22:1--22:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.22},
  URN =		{urn:nbn:de:0030-drops-111437},
  doi =		{10.4230/LIPIcs.ESA.2019.22},
  annote =	{Keywords: Randomized incremental construction, Delaunay triangulations, Voronoi diagrams, polyhedral surfaces, probabilistic analysis}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.25
Kernelization of the Subset General Position Problem in Geometry

Authors: Jean-Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, and Sudeshna Kolay

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in \mathbb{R}^d and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in general position with respect to a subsystem of hyperplanes in \mathbb{R}^d if no d+1 points lie on the same hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations. When parameterized by k then we exhibit a polynomial kernelization for the problem. When parameterized by h=n-k, or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic assumptions. We can also exhibit similar kernelization results for d-Polynomial Subset General Position, where a vector space of polynomials of degree at most d are specified as the underlying subsystem such that the size of the basis for this vector space is b. The objective is to find a set of at least k input points, or in the dual delete at most h = n-k points, such that no b+1 points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position. We also study general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem.
Cite as

Jean-Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, and Sudeshna Kolay. Kernelization of the Subset General Position Problem in Geometry. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{boissonnat_et_al:LIPIcs.MFCS.2017.25,
  author =	{Boissonnat, Jean-Daniel and Dutta, Kunal and Ghosh, Arijit and Kolay, Sudeshna},
  title =	{{Kernelization of the Subset General Position Problem in Geometry}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{25:1--25:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.25},
  URN =		{urn:nbn:de:0030-drops-80863},
  doi =		{10.4230/LIPIcs.MFCS.2017.25},
  annote =	{Keywords: Incidence Geometry, Kernel Lower bounds, Hyperplanes, Bounded degree polynomials}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.38
Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

Authors: Kunal Dutta, Arijit Ghosh, Bruno Jartoux, and Nabil H. Mustafa

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity. In this paper we present several new results and applications related to packings: * an optimal lower bound for shallow packings, * improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry, * we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted epsilon-net results follow immediately, and * simplifying and generalizing one of the main technical tools in [Fox et al. , J. of the EMS, to appear].
Cite as

Kunal Dutta, Arijit Ghosh, Bruno Jartoux, and Nabil H. Mustafa. Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dutta_et_al:LIPIcs.SoCG.2017.38,
  author =	{Dutta, Kunal and Ghosh, Arijit and Jartoux, Bruno and Mustafa, Nabil H.},
  title =	{{Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.38},
  URN =		{urn:nbn:de:0030-drops-71991},
  doi =		{10.4230/LIPIcs.SoCG.2017.38},
  annote =	{Keywords: Epsilon-nets, Haussler's packing lemma, Mnets, shallow-cell complexity, shallow packing lemma}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.96
Two Proofs for Shallow Packings

Authors: Kunal Dutta, Esther Ezra, and Arijit Ghosh

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

Abstract
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A delta-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v in W is greater than delta, where delta > 0 is an integer parameter. The delta-packing number is then defined as the cardinality of the largest delta-separated subcollection of V. Haussler showed an asymptotically tight bound of Theta((n / delta)^d) on the delta-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X' of X of size m <= n and for any parameter 1 <= k <= m, the number of vectors of length at most k in the restriction of V to X' is only O(m^{d_1} k^{d-d_1}), for a fixed integer d > 0 and a real parameter 1 <= d_1 <= d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = d). In this case when V is "k-shallow" (all vector lengths are at most k), we show that its delta-packing number is O(n^{d_1} k^{d-d_1} / delta^d), matching Haussler's bound for the special cases where d_1=d or k=n. We present two proofs, the first is an extension of Haussler's approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler's proof.
Cite as

Kunal Dutta, Esther Ezra, and Arijit Ghosh. Two Proofs for Shallow Packings. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 96-110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dutta_et_al:LIPIcs.SOCG.2015.96,
  author =	{Dutta, Kunal and Ezra, Esther and Ghosh, Arijit},
  title =	{{Two Proofs for Shallow Packings}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{96--110},
  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.2015.96},
  URN =		{urn:nbn:de:0030-drops-51493},
  doi =		{10.4230/LIPIcs.SOCG.2015.96},
  annote =	{Keywords: Set systems of bounded primal shatter dimension, delta-packing \& Haussler’s approach, relative approximations, Clarkson-Shor random sampling approach}
}
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