4 Search Results for "Ram�rez, Sergio"


Document
Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets

Authors: Boris Aronov, Esther Ezra, and Micha Sharir

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


Abstract
We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in A×B×C, a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly O(n^(28/15)) constant-degree polynomial sign tests, for the special case where two of the sets lie on one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to any other problem where we seek a triple that satisfies a single polynomial equation, e.g., determining whether A× B× C contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. [Luis Barba et al., 2019] and by Chan [Timothy M. Chan, 2020], where all three sets A, B, and C are assumed to be one-dimensional. While there are common features in the high-level approaches, here and in [Luis Barba et al., 2019], the actual analysis in this work becomes more involved and requires new methods and techniques, involving polynomial partitions and other related tools. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple (a,b,c) ∈ A×B×C that simultaneously satisfies two real polynomial equations. For example, this is the setup when testing for the existence of pairs of similar triangles spanned by the input points, in various contexts discussed later in the paper. We show that problems of this kind can be solved with roughly O(n^(24/13)) constant-degree polynomial sign tests. These problems can be extended to higher dimensions in various ways, and we present subquadratic solutions to some of these extensions, in the algebraic decision-tree model.

Cite as

Boris Aronov, Esther Ezra, and Micha Sharir. Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Copy BibTex To Clipboard

@InProceedings{aronov_et_al:LIPIcs.SoCG.2020.8,
  author =	{Aronov, Boris and Ezra, Esther and Sharir, Micha},
  title =	{{Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{8:1--8:14},
  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.2020.8},
  URN =		{urn:nbn:de:0030-drops-121666},
  doi =		{10.4230/LIPIcs.SoCG.2020.8},
  annote =	{Keywords: Algebraic decision tree, Polynomial partition, Collinearity testing, 3SUM-hard problems, Polynomials vanishing on Cartesian products}
}
Document
Further Results on Colored Range Searching

Authors: Timothy M. Chan, Qizheng He, and Yakov Nekrich

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


Abstract
We present a number of new results about range searching for colored (or "categorical") data: 1) For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,…,n}. Previous data structures require O((log n)/(log log n) + k) query time. Our result also implies improvements in higher constant dimensions. 2) Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log²n) query time. 3) For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored "type-2" range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O((log n)/(log log n) + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k (log n)/(log log n)) time. Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.

Cite as

Timothy M. Chan, Qizheng He, and Yakov Nekrich. Further Results on Colored Range Searching. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Copy BibTex To Clipboard

@InProceedings{chan_et_al:LIPIcs.SoCG.2020.28,
  author =	{Chan, Timothy M. and He, Qizheng and Nekrich, Yakov},
  title =	{{Further Results on Colored Range Searching}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{28:1--28:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.28},
  URN =		{urn:nbn:de:0030-drops-121868},
  doi =		{10.4230/LIPIcs.SoCG.2020.28},
  annote =	{Keywords: Range searching, geometric data structures, randomized incremental construction, random sampling, word RAM}
}
Document
Four-Dimensional Dominance Range Reporting in Linear Space

Authors: Yakov Nekrich

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


Abstract
In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in O(log^{1+ε} n + k log^ε n) time, where k is the number of reported points, n is the number of points in the data structure, and ε is an arbitrarily small positive constant. Our second data structure uses O(n log^ε n) space and answers queries in O(log n+k) time. These are the first data structures for this problem that use linear (resp. O(n log^ε n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^ε n)-space data structure supports queries in O(n^ε + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ≥ 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n (log n/log log n)^{d-3} + k) time using O(n log^{d-4+ε} n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time.

Cite as

Yakov Nekrich. Four-Dimensional Dominance Range Reporting in Linear Space. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Copy BibTex To Clipboard

@InProceedings{nekrich:LIPIcs.SoCG.2020.59,
  author =	{Nekrich, Yakov},
  title =	{{Four-Dimensional Dominance Range Reporting in Linear Space}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{59:1--59:14},
  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.2020.59},
  URN =		{urn:nbn:de:0030-drops-122170},
  doi =		{10.4230/LIPIcs.SoCG.2020.59},
  annote =	{Keywords: Range searching, geometric data structures, word RAM}
}
Document
Reasoning About Distributed Knowledge of Groups with Infinitely Many Agents

Authors: Michell Guzmán, Sophia Knight, Santiago Quintero, Sergio Ramírez, Camilo Rueda, and Frank Valencia

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)


Abstract
Spatial constraint systems (scs) are semantic structures for reasoning about spatial and epistemic information in concurrent systems. We develop the theory of scs to reason about the distributed information of potentially infinite groups. We characterize the notion of distributed information of a group of agents as the infimum of the set of join-preserving functions that represent the spaces of the agents in the group. We provide an alternative characterization of this notion as the greatest family of join-preserving functions that satisfy certain basic properties. We show compositionality results for these characterizations and conditions under which information that can be obtained by an infinite group can also be obtained by a finite group. Finally, we provide algorithms that compute the distributive group information of finite groups.

Cite as

Michell Guzmán, Sophia Knight, Santiago Quintero, Sergio Ramírez, Camilo Rueda, and Frank Valencia. Reasoning About Distributed Knowledge of Groups with Infinitely Many Agents. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Copy BibTex To Clipboard

@InProceedings{guzman_et_al:LIPIcs.CONCUR.2019.29,
  author =	{Guzm\'{a}n, Michell and Knight, Sophia and Quintero, Santiago and Ram{\'\i}rez, Sergio and Rueda, Camilo and Valencia, Frank},
  title =	{{Reasoning About Distributed Knowledge of Groups with Infinitely Many Agents}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.29},
  URN =		{urn:nbn:de:0030-drops-109314},
  doi =		{10.4230/LIPIcs.CONCUR.2019.29},
  annote =	{Keywords: Reasoning about Groups, Distributed Knowledge, Infinitely Many Agents, Reasoning about Space, Algebraic Modeling}
}
  • Refine by Author
  • 2 Nekrich, Yakov
  • 1 Aronov, Boris
  • 1 Chan, Timothy M.
  • 1 Ezra, Esther
  • 1 Guzmán, Michell
  • Show More...

  • Refine by Classification
  • 3 Theory of computation → Computational geometry
  • 2 Theory of computation → Data structures design and analysis
  • 1 Theory of computation
  • 1 Theory of computation → Concurrency
  • 1 Theory of computation → Distributed computing models
  • Show More...

  • Refine by Keyword
  • 2 Range searching
  • 2 geometric data structures
  • 2 word RAM
  • 1 3SUM-hard problems
  • 1 Algebraic Modeling
  • Show More...

  • Refine by Type
  • 4 document

  • Refine by Publication Year
  • 3 2020
  • 1 2019

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail