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Documents authored by Kumar, Nirman

Document
DOI: 10.4230/LIPIcs.ISAAC.2021.41
Separated Red Blue Center Clustering

Authors: Marzieh Eskandari, Bhavika Khare, and Nirman Kumar

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
We study a generalization of k-center clustering, first introduced by Kavand et. al., where instead of one set of centers, we have two types of centers, p red and q blue, and where each red center is at least α distant from each blue center. The goal is to minimize the covering radius. We provide an approximation algorithm for this problem, and a polynomial-time algorithm for the constrained problem, where all the centers must lie on a line 𝓁.
Cite as

Marzieh Eskandari, Bhavika Khare, and Nirman Kumar. Separated Red Blue Center Clustering. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{eskandari_et_al:LIPIcs.ISAAC.2021.41,
  author =	{Eskandari, Marzieh and Khare, Bhavika and Kumar, Nirman},
  title =	{{Separated Red Blue Center Clustering}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{41:1--41:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.41},
  URN =		{urn:nbn:de:0030-drops-154740},
  doi =		{10.4230/LIPIcs.ISAAC.2021.41},
  annote =	{Keywords: Algorithms, Facility Location, Clustering, Approximation Algorithms, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2019.10
Approximating Distance Measures for the Skyline

Authors: Nirman Kumar, Benjamin Raichel, Stavros Sintos, and Gregory Van Buskirk

Published in: LIPIcs, Volume 127, 22nd International Conference on Database Theory (ICDT 2019)

Abstract
In multi-parameter decision making, data is usually modeled as a set of points whose dimension is the number of parameters, and the skyline or Pareto points represent the possible optimal solutions for various optimization problems. The structure and computation of such points have been well studied, particularly in the database community. As the skyline can be quite large in high dimensions, one often seeks a compact summary. In particular, for a given integer parameter k, a subset of k points is desired which best approximates the skyline under some measure. Various measures have been proposed, but they mostly treat the skyline as a discrete object. By viewing the skyline as a continuous geometric hull, we propose a new measure that evaluates the quality of a subset by the Hausdorff distance of its hull to the full hull. We argue that in many ways our measure more naturally captures what it means to approximate the skyline. For our new geometric skyline approximation measure, we provide a plethora of results. Specifically, we provide (1) a near linear time exact algorithm in two dimensions, (2) APX-hardness results for dimensions three and higher, (3) approximation algorithms for related variants of our problem, and (4) a practical and efficient heuristic which uses our geometric insights into the problem, as well as various experimental results to show the efficacy of our approach.
Cite as

Nirman Kumar, Benjamin Raichel, Stavros Sintos, and Gregory Van Buskirk. Approximating Distance Measures for the Skyline. In 22nd International Conference on Database Theory (ICDT 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 127, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kumar_et_al:LIPIcs.ICDT.2019.10,
  author =	{Kumar, Nirman and Raichel, Benjamin and Sintos, Stavros and Van Buskirk, Gregory},
  title =	{{Approximating Distance Measures for the Skyline}},
  booktitle =	{22nd International Conference on Database Theory (ICDT 2019)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-101-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{127},
  editor =	{Barcelo, Pablo and Calautti, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2019.10},
  URN =		{urn:nbn:de:0030-drops-103125},
  doi =		{10.4230/LIPIcs.ICDT.2019.10},
  annote =	{Keywords: Skyline, Pareto optimal, Approximation, Hardness, Multi-criteria decision making}
}
Document
DOI: 10.4230/LIPIcs.SEA.2017.7
Efficient Algorithms for k-Regret Minimizing Sets

Authors: Pankaj K. Agarwal, Nirman Kumar, Stavros Sintos, and Subhash Suri

Published in: LIPIcs, Volume 75, 16th International Symposium on Experimental Algorithms (SEA 2017)

Abstract
A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item's attributes with a user's weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with respect to all possible user weight vectors. We show that k-regret minimization is NP-Complete for all dimensions d>=3, settling an open problem from Chester et al. [VLDB 2014]. Our main algorithmic contributions are two approximation algorithms, both with provable guarantees, one based on coresets and another based on hitting sets. We perform extensive experimental evaluation of our algorithms, using both real-world and synthetic data, and compare their performance against the solution proposed in [VLDB 14]. The results show that our algorithms are significantly faster and scalable to much larger sets than the greedy algorithm of Chester et al. for comparable quality answers.
Cite as

Pankaj K. Agarwal, Nirman Kumar, Stavros Sintos, and Subhash Suri. Efficient Algorithms for k-Regret Minimizing Sets. In 16th International Symposium on Experimental Algorithms (SEA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 75, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agarwal_et_al:LIPIcs.SEA.2017.7,
  author =	{Agarwal, Pankaj K. and Kumar, Nirman and Sintos, Stavros and Suri, Subhash},
  title =	{{Efficient Algorithms for k-Regret Minimizing Sets}},
  booktitle =	{16th International Symposium on Experimental Algorithms (SEA 2017)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-036-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{75},
  editor =	{Iliopoulos, Costas S. and Pissis, Solon P. and Puglisi, Simon J. and Raman, Rajeev},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2017.7},
  URN =		{urn:nbn:de:0030-drops-76321},
  doi =		{10.4230/LIPIcs.SEA.2017.7},
  annote =	{Keywords: regret minimizing sets, skyline, top-k query, coreset, hitting set}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.31
Most Likely Voronoi Diagrams in Higher Dimensions

Authors: Nirman Kumar, Benjamin Raichel, Subhash Suri, and Kevin Verbeek

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
The Most Likely Voronoi Diagram is a generalization of the well known Voronoi Diagrams to a stochastic setting, where a stochastic point is a point associated with a given probability of existence, and the cell for such a point is the set of points which would classify the given point as its most likely nearest neighbor. We investigate the complexity of this subdivision of space in d dimensions. We show that in the general case, the complexity of such a subdivision is Omega(n^{2d}) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek, which first defined the Most Likely Voronoi Diagram. We also show that when the probabilities are assigned using a random permutation of a fixed set of values, in expectation the complexity is only ~O(n^{ceil{d/2}}) where the ~O(*) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors.
Cite as

Nirman Kumar, Benjamin Raichel, Subhash Suri, and Kevin Verbeek. Most Likely Voronoi Diagrams in Higher Dimensions. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2016.31,
  author =	{Kumar, Nirman and Raichel, Benjamin and Suri, Subhash and Verbeek, Kevin},
  title =	{{Most Likely Voronoi Diagrams in Higher Dimensions}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.31},
  URN =		{urn:nbn:de:0030-drops-68667},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.31},
  annote =	{Keywords: Uncertainty, Lower bounds, Voronoi Diagrams, Stochastic}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.38
Hyperplane Separability and Convexity of Probabilistic Point Sets

Authors: Martin Fink, John Hershberger, Nirman Kumar, and Subhash Suri

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

Abstract
We describe an O(n^d) time algorithm for computing the exact probability that two d-dimensional probabilistic point sets are linearly separable, for any fixed d >= 2. A probabilistic point in d-space is the usual point, but with an associated (independent) probability of existence. We also show that the d-dimensional separability problem is equivalent to a (d+1)-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of n probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership by a factor of n [Agarwal et al., ESA, 2014]. In addition, our algorithms can handle "input degeneracies" in which more than k+1 points may lie on a k-dimensional subspace, thus resolving an open problem in [Agarwal et al., ESA, 2014]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n^2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal.
Cite as

Martin Fink, John Hershberger, Nirman Kumar, and Subhash Suri. Hyperplane Separability and Convexity of Probabilistic Point Sets. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fink_et_al:LIPIcs.SoCG.2016.38,
  author =	{Fink, Martin and Hershberger, John and Kumar, Nirman and Suri, Subhash},
  title =	{{Hyperplane Separability and Convexity of Probabilistic Point Sets}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{38:1--38:16},
  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.2016.38},
  URN =		{urn:nbn:de:0030-drops-59305},
  doi =		{10.4230/LIPIcs.SoCG.2016.38},
  annote =	{Keywords: probabilistic separability, uncertain data, 3-SUM hardness, topological sweep, hyperplane separation, multi-dimensional data}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.374
Space Exploration via Proximity Search

Authors: Sariel Har-Peled, Nirman Kumar, David M. Mount, and Benjamin Raichel

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2014.315
Robust Proximity Search for Balls Using Sublinear Space

Authors: Sariel Har-Peled and Nirman Kumar

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

Abstract
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.
Cite as

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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