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

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.

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} }

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

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.

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} }

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

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.

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} }

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

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.

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

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

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].

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} }

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

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.

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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