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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Let P be a set of m points in ℝ², let Σ be a set of n semi-algebraic sets of constant complexity in ℝ², let (S,+) be a semigroup, and let w: P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P∩σ) for every σ ∈ Σ in overall expected time O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^*(⋅) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an on-line manner. The latter takes O^*(m^{s/(2s-1)}n^{(2s-2)/(2s-1)} + m + n) time.
Let Φ: Σ × P → {0,1} be the Boolean predicate (of constant complexity) such that Φ(σ,p) = 1 if p ∈ σ and 0 otherwise, and let Σ_Φ P = {(σ,p) ∈ Σ× P ∣ Φ(σ,p) = 1}. Our algorithm actually computes a partition ℬ_Φ of Σ_Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n). It is straightforward to compute w(P∩σ) for all σ ∈ Σ from ℬ_Φ. Similarly, if η: Σ → S is a weight function on the regions of Σ, ∑_{σ ∈ Σ: p ∈ σ} η(σ), for every point p ∈ P, can be computed from ℬ_Φ in a straightforward manner. We also mention a few other applications of computing ℬ_Φ.

Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2024.4, author = {Agarwal, Pankaj K. and Ezra, Esther and Sharir, Micha}, title = {{Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {4:1--4:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.4}, URN = {urn:nbn:de:0030-drops-199497}, doi = {10.4230/LIPIcs.SoCG.2024.4}, annote = {Keywords: Range-searching, semi-algebraic sets, pseudo-lines, duality, geometric cuttings} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Let ℬ be a set of n unit balls in ℝ³. We present a linear-size data structure for storing ℬ that can determine in O^*(n^{1/2}) time whether a query line intersects any ball of ℬ and report all k such balls in additional O(k) time. The data structure can be constructed in O(n log n) time. (The O^*(⋅) notation hides subpolynomial factors, e.g., of the form O(n^ε), for arbitrarily small ε > 0, and their coefficients which depend on ε.)
We also consider the dual problem: Let ℒ be a set of n lines in ℝ³. We preprocess ℒ, in O^*(n²) time, into a data structure of size O^*(n²) that can determine in O^*(1) time whether a query unit ball intersects any line of ℒ, or report all k such lines in additional O(k) time.

Pankaj K. Agarwal and Esther Ezra. Line Intersection Searching Amid Unit Balls in 3-Space. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.5, author = {Agarwal, Pankaj K. and Ezra, Esther}, title = {{Line Intersection Searching Amid Unit Balls in 3-Space}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.5}, URN = {urn:nbn:de:0030-drops-178559}, doi = {10.4230/LIPIcs.SoCG.2023.5}, annote = {Keywords: Intersection searching, cylindrical range searching, partition trees, union of cylinders} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We develop data structures for intersection detection queries in four dimensions that involve segments, triangles and tetrahedra. Specifically, we study two main problems: (i) Preprocess a set of n tetrahedra in {ℝ}⁴ into a data structure for answering segment-intersection queries amid the given tetrahedra (referred to as segment-tetrahedron intersection queries), and (ii) Preprocess a set of n triangles in {ℝ}⁴ into a data structure that supports triangle-intersection queries amid the input triangles (referred to as triangle-triangle intersection queries). As far as we can tell, these problems have not been previously studied.
For problem (i), we first present a "standard" solution which, for any prespecified value n ≤ s ≤ n⁶ of a so-called storage parameter s, yields a data structure with O^*(s) storage and expected preprocessing, which answers an intersection query in O^*(n/s^{1/6}) time (here and in what follows, the O^*(⋅) notation hides subpolynomial factors). For problem (ii), using similar arguments, we present a solution that has the same asymptotic performance bounds.
We then improve the solution for problem (i), and present a more intricate data structure that uses O^*(n²) storage and expected preprocessing, and answers a segment-tetrahedron intersection query in O^*(n^{1/2}) time. Using the parametric search technique of Agarwal and Matoušek [P. K. Agarwal and J. Matoušek, 1993], we can obtain data structures with similar performance bounds for the ray-shooting problem amid tetrahedra in {ℝ}⁴. Unfortunately, so far we do not know how to obtain a similar improvement for problem (ii).
Our algorithms are based on a primal-dual technique for range searching with semi-algebraic sets, based on recent advances in this area [P. K. Agarwal et al., 2021; J. Matoušek and Z. Patáková, 2015]. As this is a result of independent interest, we spell out the details of this technique.
As an application, we present a solution to the problem of "continuous collision detection" amid moving tetrahedra in 3-space. That is, the workspace consists of n tetrahedra, each moving at its own fixed velocity, and the goal is to detect a collision between some pair of moving tetrahedra. Using our solutions to problems (i) and (ii), we obtain an algorithm that detects a collision in O^*(n^{12/7}) expected time. We also present further applications, including an output-sensitive algorithm for constructing the arrangement of n tetrahedra in ℝ⁴ and an output-sensitive algorithm for constructing the intersection or union of two or several nonconvex polyhedra in ℝ⁴.

Esther Ezra and Micha Sharir. Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ezra_et_al:LIPIcs.ESA.2022.51, author = {Ezra, Esther and Sharir, Micha}, title = {{Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {51:1--51:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.51}, URN = {urn:nbn:de:0030-drops-169895}, doi = {10.4230/LIPIcs.ESA.2022.51}, annote = {Keywords: Computational geometry, Ray shooting, Tetrahedra in \{\mathbb{R}\}⁴, Intersection queries in \{\mathbb{R}\}⁴, Polynomial partitioning, Range searching, Semi-algebraic sets, Tradeoff} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Let 𝒯 be a set of n planar semi-algebraic regions in ℝ³ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess 𝒯 into a data structure so that for a query object γ, which is also a plate, we can quickly answer various intersection queries, such as detecting whether γ intersects any plate of 𝒯, reporting all the plates intersected by γ, or counting them. We focus on two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in ℝ³ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in ℝ³. These interesting special cases form the building blocks for the general case.
By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if 𝒯 is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^{4/3}) storage (where the O^*(⋅) notation hides subpolynomial factors) and answers an intersection query in O^*(n^{2/3}) time. Alternatively, by increasing the storage to O^*(n^{3/2}), the query time can be decreased to O^*(n^{ρ}), where ρ = (2t-3)/3(t-1) < 2/3 and t ≥ 3 is the number of parameters needed to represent the query arcs.

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2022.4, author = {Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Katz, Matthew J. and Sharir, Micha}, title = {{Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.4}, URN = {urn:nbn:de:0030-drops-160126}, doi = {10.4230/LIPIcs.SoCG.2022.4}, annote = {Keywords: Intersection searching, Semi-algebraic range searching, Point-enclosure queries, Ray-shooting queries, Polynomial partitions, Cylindrical algebraic decomposition, Multi-level partition trees, Collision detection} }

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

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ ∈ C, the number of intersection points between the segments of A and those of B that lie in Δ. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n²/log²n) log^O(1) log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n^{60/31+ε}), for any ε > 0.
Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020).
A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.

Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir. Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{aronov_et_al:LIPIcs.ISAAC.2021.3, author = {Aronov, Boris and de Berg, Mark and Cardinal, Jean and Ezra, Esther and Iacono, John and Sharir, Micha}, title = {{Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {3:1--3:15}, 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.2021.3}, URN = {urn:nbn:de:0030-drops-154363}, doi = {10.4230/LIPIcs.ISAAC.2021.3}, annote = {Keywords: Computational geometry, Algebraic decision-tree model, Polynomial partitioning, Primal-dual range searching, Order types, Point location, Hierarchical partitions} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in ℝ³, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in ℝ³, and (v) output-sensitive construction of an arrangement of triangles in three dimensions.
Our approach is based on the polynomial partitioning technique.
For example, our ray-shooting algorithm processes a set of n triangles in ℝ³ into a data structure for answering ray shooting queries amid the given triangles, which uses O(n^{3/2+ε}) storage and preprocessing, and answers a query in O(n^{1/2+ε}) time, for any ε > 0. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly n^{5/8}. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results.
We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer m queries on n objects in max{O(m^{2/3}n^{5/6+{ε}} + n^{1+ε}), O(m^{5/6+ε}n^{2/3} + m^{1+ε})} time, for any ε > 0, again an improvement over the earlier bounds.

Esther Ezra and Micha Sharir. On Ray Shooting for Triangles in 3-Space and Related Problems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ezra_et_al:LIPIcs.SoCG.2021.34, author = {Ezra, Esther and Sharir, Micha}, title = {{On Ray Shooting for Triangles in 3-Space and Related Problems}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.34}, URN = {urn:nbn:de:0030-drops-138332}, doi = {10.4230/LIPIcs.SoCG.2021.34}, annote = {Keywords: Ray shooting, Three dimensions, Polynomial partitioning, Tradeoff} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We show that the maximum number of pairwise non-overlapping k-rich lenses (lenses formed by at least k circles) in an arrangement of n circles in the plane is O(n^{3/2}log(n / k^3) k^{-5/2} + n/k), and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is O(n^{3/2}log(n/k^3) k^{-3/2} + n). Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of Agarwal et al. (JACM 2004) and Marcus and Tardos (JCTA 2006) on the number of point-circle incidences in the plane. Extensions to families of more general algebraic curves and some other related problems are also considered.

Esther Ezra, Orit E. Raz, Micha Sharir, and Joshua Zahl. On Rich Lenses in Planar Arrangements of Circles and Related Problems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ezra_et_al:LIPIcs.SoCG.2021.35, author = {Ezra, Esther and Raz, Orit E. and Sharir, Micha and Zahl, Joshua}, title = {{On Rich Lenses in Planar Arrangements of Circles and Related Problems}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {35:1--35:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.35}, URN = {urn:nbn:de:0030-drops-138343}, doi = {10.4230/LIPIcs.SoCG.2021.35}, annote = {Keywords: Lenses, Circles, Polynomial partitioning, Incidences} }

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

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.

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.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} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in R^d and if D >= 1 is an integer, then there is a d-variate polynomial P of degree at most D so that each connected component of R^d \ Z(P) intersects O(n/D^{d-g}) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently - the expected running time of our algorithm is linear in |S|. Our approach exploits the technique of quantifier elimination combined with that of epsilon-samples.
We present four applications of our result. The first is a data structure for answering point-enclosure queries among a family of semi-algebraic sets in R^d in O(log n) time, with storage complexity and expected preprocessing time of O(n^{d+epsilon}). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(n^{t+epsilon}) storage and expected preprocessing time, where t > 0 is an integer that depends on d and the description complexity of the ranges. The third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in R^{d} in O(log^2 n) time, with O(n^{d+epsilon}) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.5, author = {Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Zahl, Joshua}, title = {{An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.5}, URN = {urn:nbn:de:0030-drops-104096}, doi = {10.4230/LIPIcs.SoCG.2019.5}, annote = {Keywords: Polynomial partitioning, quantifier elimination, semi-algebraic range spaces, epsilon-samples} }

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

We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^2 log^2 n),improving the recent bound O(n^3 log^3 n) of Cardinal et al. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional Knapsack or SubsetSum problems is only O(n^3 log n), improving the currently best known bounds by a factor of n. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions.
A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements.

Esther Ezra and Micha Sharir. A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ezra_et_al:LIPIcs.SoCG.2017.41, author = {Ezra, Esther and Sharir, Micha}, title = {{A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {41:1--41:15}, 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.2017.41}, URN = {urn:nbn:de:0030-drops-71853}, doi = {10.4230/LIPIcs.SoCG.2017.41}, annote = {Keywords: k-SUM and k-LDT, linear decision tree, hyperplane arrangements, point-location, vertical decompositions, Epsilon-cuttings} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Sigma), where each element x in X lies in t randomly selected sets of Sigma, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |\Sigma| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.

Esther Ezra and Shachar Lovett. On the Beck-Fiala Conjecture for Random Set Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 29:1-29:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ezra_et_al:LIPIcs.APPROX-RANDOM.2016.29, author = {Ezra, Esther and Lovett, Shachar}, title = {{On the Beck-Fiala Conjecture for Random Set Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {29:1--29:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.29}, URN = {urn:nbn:de:0030-drops-66526}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.29}, annote = {Keywords: Discrepancy theory, Beck-Fiala conjecture, Random set systems} }

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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} }