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Documents authored by Ooms, Aurélien

Document
DOI: 10.4230/LIPIcs.SWAT.2020.20
Sparse Regression via Range Counting

Authors: Jean Cardinal and Aurélien Ooms

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

Abstract
The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set S of n points in ℝ^d, a point y∈ ℝ^d, and an integer 2 ≤ k ≤ d, find an affine combination of at most k points of S that is nearest to y. We describe a O(n^{k-1} log^{d-k+2} n)-time randomized (1+ε)-approximation algorithm for this problem with d and ε constant. This is the first algorithm for this problem running in time o(n^k). Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP'18), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where k = d = O(1), we provide a simple O_δ(n^{d-1+δ})-time deterministic exact algorithm, for any δ > 0. Finally, we show how to adapt the approximation algorithm for the sparse linear regression and sparse convex regression problems with the same running time, up to polylogarithmic factors.
Cite as

Jean Cardinal and Aurélien Ooms. Sparse Regression via Range Counting. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cardinal_et_al:LIPIcs.SWAT.2020.20,
  author =	{Cardinal, Jean and Ooms, Aur\'{e}lien},
  title =	{{Sparse Regression via Range Counting}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-150-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{162},
  editor =	{Albers, Susanne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.20},
  URN =		{urn:nbn:de:0030-drops-122677},
  doi =		{10.4230/LIPIcs.SWAT.2020.20},
  annote =	{Keywords: Sparse Linear Regression, Orthogonal Range Searching, Affine Degeneracy Testing, Nearest Neighbors, Hyperplane Arrangements}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.58
Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

Authors: Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André van Renssen, and Marcel Roeloffzen

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

Abstract
We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min(n^omega, n^2 + nh log h + chi^2)) time, where omega<2.373 denotes the matrix multiplication exponent and chi in Omega(n) cap O(n^2) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n^2 log n) time.
Cite as

Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André van Renssen, and Marcel Roeloffzen. Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arseneva_et_al:LIPIcs.ISAAC.2018.58,
  author =	{Arseneva, Elena and Chiu, Man-Kwun and Korman, Matias and Markovic, Aleksandar and Okamoto, Yoshio and Ooms, Aur\'{e}lien and van Renssen, Andr\'{e} and Roeloffzen, Marcel},
  title =	{{Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{58:1--58:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.58},
  URN =		{urn:nbn:de:0030-drops-100060},
  doi =		{10.4230/LIPIcs.ISAAC.2018.58},
  annote =	{Keywords: Rectilinear link distance, polygonal domain, diameter, radius}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.20
Subquadratic Encodings for Point Configurations

Authors: Jean Cardinal, Timothy M. Chan, John Iacono, Stefan Langerman, and Aurélien Ooms

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.
Cite as

Jean Cardinal, Timothy M. Chan, John Iacono, Stefan Langerman, and Aurélien Ooms. Subquadratic Encodings for Point Configurations. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cardinal_et_al:LIPIcs.SoCG.2018.20,
  author =	{Cardinal, Jean and Chan, Timothy M. and Iacono, John and Langerman, Stefan and Ooms, Aur\'{e}lien},
  title =	{{Subquadratic Encodings for Point Configurations}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.20},
  URN =		{urn:nbn:de:0030-drops-87337},
  doi =		{10.4230/LIPIcs.SoCG.2018.20},
  annote =	{Keywords: point configuration, order type, chirotope, succinct data structure}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2018.14
Improved Bounds for Guarding Plane Graphs with Edges

Authors: Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, and Sander Verdonschot

Published in: LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Abstract
An edge guard set of a plane graph G is a subset Gamma of edges of G such that each face of G is incident to an endpoint of an edge in Gamma. Such a set is said to guard G. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G: 1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (1997) that G can be guarded with at most 2n/5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n/8 edges for any plane graph. 2) We prove that there exists an edge guard set of G with at most n/(3) + alpha/9 edges, where alpha is the number of quadrilateral faces in G. This improves the previous bound of n/(3) + alpha by Bose, Kirkpatrick, and Li (2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that n/(3) edges suffice, removing the dependence on alpha.
Cite as

Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, and Sander Verdonschot. Improved Bounds for Guarding Plane Graphs with Edges. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biniaz_et_al:LIPIcs.SWAT.2018.14,
  author =	{Biniaz, Ahmad and Bose, Prosenjit and Ooms, Aur\'{e}lien and Verdonschot, Sander},
  title =	{{Improved Bounds for Guarding Plane Graphs with Edges}},
  booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
  pages =	{14:1--14:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-068-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{101},
  editor =	{Eppstein, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.14},
  URN =		{urn:nbn:de:0030-drops-88403},
  doi =		{10.4230/LIPIcs.SWAT.2018.14},
  annote =	{Keywords: edge guards, graph coloring, four-color theorem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.13
Subquadratic Algorithms for Algebraic Generalizations of 3SUM

Authors: Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon

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

Abstract
The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11/6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given n points in the plane, do three of them lie on a line?", a key problem in computational geometry. We prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+e}) that solve 3POL, and that 3POL can be solved in O(n^2 (log log n)^{3/2} / (log n)^{1/2}) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on o((log n)^{1/6}/(log log n)^{1/2}) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.
Cite as

Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon. Subquadratic Algorithms for Algebraic Generalizations of 3SUM. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barba_et_al:LIPIcs.SoCG.2017.13,
  author =	{Barba, Luis and Cardinal, Jean and Iacono, John and Langerman, Stefan and Ooms, Aur\'{e}lien and Solomon, Noam},
  title =	{{Subquadratic Algorithms for Algebraic Generalizations of 3SUM}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{13:1--13:15},
  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.2017.13},
  URN =		{urn:nbn:de:0030-drops-72214},
  doi =		{10.4230/LIPIcs.SoCG.2017.13},
  annote =	{Keywords: 3SUM, subquadratic algorithms, general position testing, range searching, dominance reporting, polynomial curves}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.25
Solving k-SUM Using Few Linear Queries

Authors: Jean Cardinal, John Iacono, and Aurélien Ooms

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c<d where d is the ceiling of k/2. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n^3 log^2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in ~O(n^{d+8}) time, and performs O(n^3 log^2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n^3 log^2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-a-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth ~O(n^3) for the k-SUM problem.
Cite as

Jean Cardinal, John Iacono, and Aurélien Ooms. Solving k-SUM Using Few Linear Queries. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2016.25,
  author =	{Cardinal, Jean and Iacono, John and Ooms, Aur\'{e}lien},
  title =	{{Solving k-SUM Using Few Linear Queries}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.25},
  URN =		{urn:nbn:de:0030-drops-63763},
  doi =		{10.4230/LIPIcs.ESA.2016.25},
  annote =	{Keywords: k-SUM problem, linear decision trees, point location, \$varepsilon\$-nets}
}
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