3 Search Results for "Sheffer, Adam"


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)


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@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
Bisector Energy and Few Distinct Distances

Authors: Ben Lund, Adam Sheffer, and Frank de Zeeuw

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


Abstract
We introduce the bisector energy of an n-point set P in the real plane, defined as the number of quadruples (a,b,c,d) from P such that a and b determine the same perpendicular bisector as c and d. If no line or circle contains M(n) points of P, then we prove that the bisector energy is O(M(n)^{2/5}n^{12/5} + M(n)n^2). We also prove the lower bound M(n)n^2, which matches our upper bound when M(n) is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If P determines O(n / sqrt(log n)) distinct distances, then for any 0 < a < 1/4, either there exists a line or circle that contains n^a points of P, or there exist n^{8/5 - 12a/5} distinct lines that contain sqrt(log n) points of P. This result provides new information on a conjecture of Erdös regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains M(n) points of P, then the number of distinct perpendicular bisectors determined by P is min{M(n)^{-2/5}n^{8/5}, M(n)^{-1}n^2}). This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over the real numbers, initiated by Elekes and Ronyai.

Cite as

Ben Lund, Adam Sheffer, and Frank de Zeeuw. Bisector Energy and Few Distinct Distances. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 537-552, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{lund_et_al:LIPIcs.SOCG.2015.537,
  author =	{Lund, Ben and Sheffer, Adam and de Zeeuw, Frank},
  title =	{{Bisector Energy and Few Distinct Distances}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{537--552},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.537},
  URN =		{urn:nbn:de:0030-drops-51086},
  doi =		{10.4230/LIPIcs.SOCG.2015.537},
  annote =	{Keywords: Combinatorial geometry, distinct distances, incidence geometry}
}
Document
Bounds on the maximum multiplicity of some common geometric graphs

Authors: Adrian Dumitrescu, Andre Schulz, Adam Sheffer, and Csaba D. Toth

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)


Abstract
We obtain new lower and upper bounds for the maximum multiplicity of some weighted, and respectively non-weighted, common geometric graphs drawn on $n$ points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of $n$ points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Omega (8.65^n) different triangulations. This improves the bound Omega (8.48^n) achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We present a new lower bound of Omega(11.97^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, Omega(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.664^n) non-crossing spanning cycles. (iv) We derive exponential lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). It was known that the number of longest non-crossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(n log n) time algorithm for computing them.

Cite as

Adrian Dumitrescu, Andre Schulz, Adam Sheffer, and Csaba D. Toth. Bounds on the maximum multiplicity of some common geometric graphs. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 637-648, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{dumitrescu_et_al:LIPIcs.STACS.2011.637,
  author =	{Dumitrescu, Adrian and Schulz, Andre and Sheffer, Adam and Toth, Csaba D.},
  title =	{{Bounds on the maximum multiplicity of some common geometric graphs}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{637--648},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.637},
  URN =		{urn:nbn:de:0030-drops-30505},
  doi =		{10.4230/LIPIcs.STACS.2011.637},
  annote =	{Keywords: combinatorial geometry, matching, triangulation, spanning tree, spanning cycle, weighted structure, non-crossing condition}
}
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