arXiv subjects

Micha Sharir

Publications and source records attributed to Micha Sharir.

At least 73 records · Page 4Linked to original sources

arXiv2015

Generalizations of the Szemerédi-Trotter Theorem

We generalize the Szemerédi-Trotter incidence theorem, to bound the number of complete \emph{flags} in higher dimensions. Specifically, for each $i=0,1,\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\R^d$ or in $\C^d$, and a (complete) flag is a tuple $(f_0,f_1,\ldots,f_{d-1})$, where $f_i\in S_i$ for each $i$ and $f_i\subset f_{i+1}$ for each $i=0,1,\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\R^3$ such that among the lines incident to a point, at most $O(1)$ of them can be coplanar, (ii) incidences with Legendrian lines in $\R^3$, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in $\R^3$ (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.

math.CO↗

arXiv2015

Distinct distances between points and lines

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $Ω(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the plane, not all on a line, the number of distances between these points and the lines that they span is $Ω(m^{4/3})$. The problem of bounding the number of distinct point-line distances can be reduced to the problem of bounding the number of tangent pairs among a finite set of lines and a finite set of circles in the plane, and we believe that this latter question is of independent interest. In the same vein, we show that $n$ circles in the plane determine at most $O(n^{3/2})$ points where two or more circles are tangent, improving the previously best known bound of $O(n^{3/2}\log n)$. Finally, we study three-dimensional versions of the distinct point-line distances problem, namely, distinct point-line distances and distinct point-plane distances. The problems studied in this paper are all new, and the bounds that we derive for them, albeit most likely not tight, are non-trivial to prove. We hope that our work will motivate further studies of these and related problems.

math.MG↗

arXiv2015

Incidences with curves in R^d

We prove that the number of incidences between $m$ points and $n$ bounded-degree curves with $k$ degrees of freedom in ${\mathbb R}^d$ is \[ I(m,n) =O\left(m^{\frac{k}{dk-d+1}+\varepsilon}n^{\frac{dk-d}{dk-d+1}}+ \sum_{j=2}^{d-1} m^{\frac{k}{jk-j+1}+\varepsilon}n^{\frac{d(j-1)(k-1)}{(d-1)(jk-j+1)}}q_j^{\frac{(d-j)(k-1)}{(d-1)(jk-j+1)}}+m+n\right), \] for any $\varepsilon>0$, where the constant of proportionality depends on $k, \varepsilon$ and $d$, provided that no $j$-dimensional surface of degree $\le c_j(k,d,\varepsilon)$, a constant parameter depending on $k$, $d$, $j$, and $\varepsilon$, contains more than $q_j$ input curves, and that the $q_j$'s satisfy certain mild conditions. This bound generalizes a recent result of Sharir and Solomon concerning point-line incidences in four dimensions (where $d=4$ and $k=2$), and partly generalizes a recent result of Guth (as well as the earlier bound of Guth and Katz) in three dimensions (Guth's three-dimensional bound has a better dependency on $q_2$). It also improves a recent $d$-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl, in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi and by Hablicsek and Scherr concerning rich lines in high-dimensional spaces.

math.CO↗

arXiv2015

Delaunay Triangulations of Degenerate Point Sets

The Delaunay triangulation (DT) is one of the most common and useful triangulations of point sets $P$ in the plane. DT is not unique when $P$ is degenerate, specifically when it contains quadruples of co-circular points. One way to achieve uniqueness is by applying a small (or infinitesimal) perturbation to $P$. We consider a specific perturbation of such degenerate sets, in which the coordinates of each point are independently perturbed by normally distributed small quantities, and investigate the effect of such perturbations on the DT of the set. We focus on two special configurations, where (1) the points of $P$ form a uniform grid, and (2) the points of $P$ are vertices of a regular polygon. We present interesting (and sometimes surprising) empirical findings and properties of the perturbed DTs for these cases, and give theoretical explanations to some of them.

cs.CG↗

arXiv2015

On the number of unit-area triangles spanned by convex grids in the plane

A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A, B\subset\mathbb R$, each of size $n^{1/2}$, the convex grid $A\times B$ spans at most $O(n^{37/17}\log^{2/17}n)$ unit-area triangles. This improves the best known upper bound $O(n^{31/14})$ recently obtained in \cite{RS}. Our analysis also applies to more general families of sets $A$, $B$, known as sets of Szemerédi--Trotter type.

math.CO↗

arXiv2015

Stable Delaunay Graphs

Let $P$ be a set of $n$ points in $\mathrm{R}^2$, and let $\mathrm{DT}(P)$ denote its Euclidean Delaunay triangulation. We introduce the notion of an edge of $\mathrm{DT}(P)$ being {\it stable}. Defined in terms of a parameter $α>0$, a Delaunay edge $pq$ is called $α$-stable, if the (equal) angles at which $p$ and $q$ see the corresponding Voronoi edge $e_{pq}$ are at least $α$. A subgraph $G$ of $\mathrm{DT}(P)$ is called {\it $(cα, α)$-stable Delaunay graph} ($\mathrm{SDG}$ in short), for some constant $c \ge 1$, if every edge in $G$ is $α$-stable and every $cα$-stable of $\mathrm{DT}(P)$ is in $G$. We show that if an edge is stable in the Euclidean Delaunay triangulation of $P$, then it is also a stable edge, though for a different value of $α$, in the Delaunay triangulation of $P$ under any convex distance function that is sufficiently close to the Euclidean norm, and vice-versa. In particular, a $6α$-stable edge in $\mathrm{DT}(P)$ is $α$-stable in the Delaunay triangulation under the distance function induced by a regular $k$-gon for $k \ge 2π/α$, and vice-versa. Exploiting this relationship and the analysis in~\cite{polydel}, we present a linear-size kinetic data structure (KDS) for maintaining an $(8α,α)$-$\mathrm{SDG}$ as the points of $P$ move. If the points move along algebraic trajectories of bounded degree, the KDS processes nearly quadratic events during the motion, each of which can processed in $O(\log n)$ time. Finally, we show that a number of useful properties of $\mathrm{DT}(P)$ are retained by $\mathrm{SDG}$ of $P$.

cs.CG↗

arXiv2015

Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited

Let $F\in\mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\subset\mathbb C$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szabó [Combinatorica, 2012], and generalizes a result of Raz, Sharir, and Solymosi [Amer. J. Math., to appear]. The same statement holds over $\mathbb{R}$, and a similar statement holds when $A, B, C$ have different sizes (with a more involved bound replacing $O(n^{11/6})$). This result provides a unified tool for improving bounds in various Erd\H os-type problems in combinatorial geometry, and we discuss several applications of this kind.

math.CO↗

arXiv2015

The number of unit-area triangles in the plane: Theme and variations

We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if $S$ consists of points on three lines, the number of unit-area triangles that $S$ spans can be $Ω(n^2)$, for any triple of lines (it is always $O(n^2)$ in this case). (ii) We show that if $S$ is a {\em convex grid} of the form $A\times B$, where $A$, $B$ are {\em convex} sets of $n^{1/2}$ real numbers each (i.e., the sequences of differences of consecutive elements of $A$ and of $B$ are both strictly increasing), then $S$ determines $O(n^{31/14})$ unit-area triangles.

math.CO↗

arXiv2015

Incidences between points and lines in R^4

We show that the number of incidences between $m$ distinct points and $n$ distinct lines in ${\mathbb R}^4$ is $O\left(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n\right)$, for a suitable absolute constant $c$, provided that no 2-plane contains more than $s$ input lines, and no hyperplane or quadric contains more than $q$ lines. The bound holds without the factor $2^{c\sqrt{\log m}}$ when $m \le n^{6/7}$ or $m \ge n^{5/3}$. Except for this factor, the bound is tight in the worst case.

math.CO↗

arXiv2015

A faster algorithm for the discrete Fréchet distance under translation

The discrete Fréchet distance is a useful similarity measure for comparing two sequences of points $P=(p_1,\ldots, p_m)$ and $Q=(q_1,\ldots,q_n)$. In many applications, the quality of the matching can be improved if we let $Q$ undergo some transformation relative to $P$. In this paper we consider the problem of finding a translation of $Q$ that brings the discrete Fréchet distance between $P$ and $Q$ to a minimum. We devise an algorithm that computes the minimum discrete Fréchet distance under translation in $\mathbb{R}^2$, and runs in $O(m^3n^2(1+\log(n/m))\log(m+n))$ time, assuming $m\leq n$. This improves a previous algorithm of Jiang et al.~\cite{JXZ08}, which runs in $O(m^3n^3 \log(m + n))$ time.

cs.CG↗

arXiv2015

Incidences between points and lines on a two-dimensional variety

We present a direct and fairly simple proof of the following incidence bound: Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${\mathbb R}^d$, for $d\ge 3$, which lie in a common algebraic two-dimensional surface of degree $D$ that does not contain any 2-flat, so that no 2-flat contains more than $s \le D$ lines of $L$. Then the number of incidences between $P$ and $L$ is $$ I(P,L)=O\left(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}\min\{n,D^{2}\}^{1/3}s^{1/3} + m + n\right). $$ When $d=3$, this improves the bound of Guth and Katz~\cite{GK2} for this special case, when $D$ is not too large. A supplementary feature of this work is a review, with detailed proofs, of several basic (and folklore) properties of ruled surfaces in three dimensions.

math.AG↗

arXiv2015

Incidences between points and lines in three dimensions

We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ O\left(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + n\right) $$ (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between $m$ and $n$). This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step in their solution of Erd{\H o}s's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth~\cite{Gu14}. The present paper presents a different and simpler derivation, with better bounds than those in \cite{Gu14}, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.

math.CO↗

arXiv2014

Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.

cs.CG↗

arXiv2014

Epsilon-Nets for Halfspaces Revisited

Given a set $P$ of $n$ points in $\mathbb{R}^3$, we show that, for any $\varepsilon >0$, there exists an $\varepsilon$-net of $P$ for halfspace ranges, of size $O(1/\varepsilon)$. We give five proofs of this result, which are arguably simpler than previous proofs \cite{msw-hnlls-90, cv-iaags-07, pr-nepen-08}. We also consider several related variants of this result, including the case of points and pseudo-disks in the plane.

cs.CG↗

arXiv2014

Sets with few distinct distances do not have heavy lines

Let $P$ be a set of $n$ points in the plane that determines at most $n/5$ distinct distances. We show that no line can contain more than $O(n^{43/52}{\rm polylog}(n))$ points of $P$. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.

math.CO↗

arXiv2014

Improved bounds for incidences between points and circles

We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension $\ge 2$, is $O*(m^{2/3}n^{2/3} + m^{6/11}n^{9/11}+m+n)$, where the $O*(\cdot)$ notation hides sub-polynomial factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in R^3 without first improving it in the plane. Nevertheless, we show that if the set of circles is required to be "truly three-dimensional" in the sense that no sphere or plane contains more than $q$ of the circles, for some $q << n$, then the bound can be improved to \[O*(m^{3/7}n^{6/7} + m^{2/3}n^{1/2}q^{1/6} + m^{6/11}n^{15/22}q^{3/22} + m + n). \] For various ranges of parameters (e.g., when $m=Θ(n)$ and $q = o(n^{7/9})$), this bound is smaller than the lower bound $Ω*(m^{2/3}n^{2/3}+m+n)$, which holds in two dimensions. We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound $O*(m^{5/11}n^{9/11} + m^{2/3}n^{1/2}q^{1/6} + m + n$. (ii) We present an improved analysis that removes the subpolynomial factors from the bound when $m=O(n^{3/2-\eps})$ for any fixed $\varepsilon >0$. (iii) We use our results to obtain the improved bound $O(m^{15/7})$ for the number of mutually similar triangles determined by any set of $m$ points in R^3. Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

math.CO↗

arXiv2014

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4nλ_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $λ_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

cs.CG↗

arXiv2014

Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=Ω(n^{4/3})$, for every pair of finite sets $A,B\subset{\mathbb R}$, with $|A|=|B|=n$ (where the constant of proportionality depends on ${\rm deg} f$), or else $f$ must be of one of the special forms $f(u,v)=h(φ(u)+ψ(v))$, or $f(u,v)=h(φ(u)\cdotψ(v))$, for some univariate polynomials $φ,ψ,h$ over ${\mathbb R}$. This significantly improves a result of Elekes and Rónyai (2000). Our results are cast in a more general form, in which we give an upper bound for the number of zeros of $z=f(x,y)$ on a triple Cartesian product $A\times B\times C$, when the sizes $|A|$, $|B|$, $|C|$ need not be the same; the upper bound is $O(n^{11/6})$ when $|A|=|B|=|C|=n$, where the constant of proportionality depends on ${\rm deg} f$, unless $f$ has one of the aforementioned special forms. This result provides a unified tool for improving bounds in various Erd\H os-type problems in geometry and additive combinatorics. Several applications of our results to problems of these kinds are presented. For example, we show that the number of distinct distances between $n$ points lying on a constant-degree parametric algebraic curve which does not contain a line, in any dimension, is $Ω(n^{4/3})$, extending the result of Pach and de Zeeuw (2013) and improving the bound of Charalambides (2012), for the special case where the curve under consideration has a polynomial parameterization. We also derive improved lower bounds for several variants of the sum-product problem in additive combinatorics.

cs.CG↗