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Micha Sharir

Publications and source records attributed to Micha Sharir.

At least 55 records · Page 3Linked to original sources
arXiv2018

Approximate Minimum-Weight Matching with Outliers under Translation

Our goal is to compare two planar point sets by finding subsets of a given size such that a minimum-weight matching between them has the smallest weight. This can be done by a translation of one set that minimizes the weight of the matching. We give efficient algorithms (a) for finding approximately optimal matchings, when the cost of a matching is the $L_p$-norm of the tuple of the Euclidean distances between the pairs of matched points, for any $p\in [1,\infty]$, and (b)~for constructing small-size approximate minimization (or matching) diagrams: partitions of the translation space into regions, together with an approximate optimal matching for each region.

cs.CG↗
arXiv2017

Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location

$\renewcommand{\Re}{\mathbb{R}}$ We re-examine parameters for the two main space decomposition techniques---bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension $d$, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension. For vertical decomposition, the combinatorial dimension is only $2d$, the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is at least $1 + d(d+1)/2$ and at most $O(d^3)$. For bottom-vertex triangulation, both the primal shatter dimension and the combinatorial dimension are $Θ(d^2)$, but there seems to be a significant gap between them, as the combinatorial dimension is $\frac12d(d+3)$, whereas the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is between $d(d+1)$ and $5d^2 \log{d}$ (for $d\ge 9$). Our main application is to point location in an arrangement of $n$ hyperplanes is $\Re^d$, in which we show that the query cost in Meiser's algorithm can be improved if one uses vertical decomposition instead of bottom-vertex triangulation, at the cost of some increase in the preprocessing cost and storage. The best query time that we can obtain is $O(d^3\log n)$, instead of $O(d^4\log d\log n)$ in Meiser's algorithm. For these bounds to hold, the preprocessing and storage are rather large (super-exponential in $d$). We discuss the tradeoff between query cost and storage (in both approaches, the one using bottom-vertex trinagulation and the one using vertical decomposition).

cs.CG↗
arXiv2017

Homotheties and incidences

We consider problems involving rich homotheties in a set S of n points in the plane (that is, homotheties that map many points of S to other points of S). By reducing these problems to incidence problems involving points and lines in R^3, we are able to obtain refined and new bounds for the number of rich homotheties, and for the number of distinct equivalence classes, under homotheties, of k-element subsets of S, for any k >= 3. We also discuss the extensions of these problems to three and higher dimensions.

cs.CG↗
arXiv2017

Dominance Product and High-Dimensional Closest Pair under $L_\infty$

Given a set $S$ of $n$ points in $\mathbb{R}^d$, the Closest Pair problem is to find a pair of distinct points in $S$ at minimum distance. When $d$ is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general $d$. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional $L_\infty$ Closest Pair problem, where $d=n^r$ for some $r > 0$, and the underlying metric is $L_\infty$. We improve and simplify previous results for $L_\infty$ Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in $O(DP(n,d)\log n)$ time, and by a randomized algorithm that runs in $O(DP(n,d))$ expected time, where $DP(n,d)$ is the time bound for computing the {\em dominance product} for $n$ points in $\mathbb{R}^d$. That is a matrix $D$, such that $D[i,j] = \bigl| \{k \mid p_i[k] \leq p_j[k]\} \bigr|$; this is the number of coordinates at which $p_j$ dominates $p_i$. For integer coordinates from some interval $[-M, M]$, we obtain an algorithm that runs in $\tilde{O}\left(\min\{Mn^{ω(1,r,1)},\, DP(n,d)\}\right)$ time, where $ω(1,r,1)$ is the exponent of multiplying an $n \times n^r$ matrix by an $n^r \times n$ matrix. We also give slightly better bounds for $DP(n,d)$, by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths).

cs.DS↗
arXiv2017

Incidences between points on a variety and planes in R^3

In this paper we establish an improved bound for the number of incidences between a set $P$ of $m$ points and a set $H$ of $n$ planes in $\mathbb R^3$, provided that the points lie on a two-dimensional nonlinear irreducible algebraic variety $V$ of constant degree. Specifically, the bound is $$ O\left( m^{2/3}n^{2/3} + m^{6/11}n^{9/11}\log^β(m^3/n) + m + n + \sum_\ell |P_\ell|\cdot |H_\ell| \right) , $$ where the constant of proportionality and the constant exponent $β$ depend on the degree of $V$, and where the sum ranges over all lines $\ell$ that are fully contained in $V$ and contain at least one point of $P$, so that, for each such $\ell$, $P_\ell = P\cap\ell$ and $H_\ell$ is the set of the planes of H that contain $\ell$. In addition, $\sum_\ell |P_\ell| = O(m)$ and $\sum_\ell |H_\ell| = O(n)$. This improves, for this special case, the earlier more general bound of Apfelbaum and Sharir (see also Brass and Knauer as well as Elekes and Tóth). This is a generalization of the incidence bound for points and circles in the plane (cf. Aronov et al., Aronov and Sharir, Marcus and Tardos), and is based on a recent result of Sharir and Zahl on the number of cuts that turn a collection of algebraic curves into pseudo-segments. The case where $V$ is a quadric is simpler to analyze, does not require the result of Sharir and Zahl, and yields the same bound as above, with $β=2/11$. We present an interesting application of our results to a problem, studied by Rudnev, on obtaining a lower bound on the number of distinct cross-ratios determined by $n$ real points, where our bound leads to a slight improvement in Rudnev's bound.

math.CO↗
arXiv2017

Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies [Guth-Zahl 2016], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for numerous incidence problems in $\mathbb R^3$. In broad terms, we consider two kinds of problems, those involving points and constant-degree algebraic \emph{curves}, and those involving points and constant-degree algebraic \emph{surfaces}. In some variants we assume that the points lie on some fixed constant-degree algebraic variety, and in others we consider arbitrary sets of points in 3-space. our results provide a "grand generalization" of most of the previous studies of (special instances of) previous works for both curves and surfaces. As an application of our point-curve incidence bound, we consider the problem of bounding the number of similar triangles spanned by a set of $n$ points in $\mathbb R^3$, and obtain the bound $O(n^{15/7}),$ thus improving the bound of Agarwal et al [2007]. As applications of our point-surface incidence bounds, we consider the problems of distinct and repeated distances determined by a set of $n$ points in $\mathbb R^3$, two of the most celebrated open problems in combinatorial geometry. We obtain new and improved bounds for two special cases, one in which the points lie on some algebraic variety of constant degree, and one involving incidences between pairs in $P_1\times P_2$, where $P_1$ is contained in a variety and $P_2$ is arbitrary.

math.CO↗
arXiv2017

Improved Bounds for 3SUM, $k$-SUM, and Linear Degeneracy

Given a set of $n$ real numbers, the 3SUM problem is to decide whether there are three of them that sum to zero. Until a recent breakthrough by Grønlund and Pettie [FOCS'14], a simple $Θ(n^2)$-time deterministic algorithm for this problem was conjectured to be optimal. Over the years many algorithmic problems have been shown to be reducible from the 3SUM problem or its variants, including the more generalized forms of the problem, such as $k$-SUM and $k$-variate linear degeneracy testing ($k$-LDT). The conjectured hardness of these problems have become extremely popular for basing conditional lower bounds for numerous algorithmic problems in P. In this paper, we show that the randomized $4$-linear decision tree complexity of 3SUM is $O(n^{3/2})$, and that the randomized $(2k-2)$-linear decision tree complexity of $k$-SUM and $k$-LDT is $O(n^{k/2})$, for any odd $k\ge 3$. These bounds improve (albeit randomized) the corresponding $O(n^{3/2}\sqrt{\log n})$ and $O(n^{k/2}\sqrt{\log n})$ decision tree bounds obtained by Grønlund and Pettie. Our technique includes a specialized randomized variant of fractional cascading data structure. Additionally, we give another deterministic algorithm for 3SUM that runs in $O(n^2 \log\log n / \log n )$ time. The latter bound matches a recent independent bound by Freund [Algorithmica 2017], but our algorithm is somewhat simpler, due to a better use of word-RAM model.

cs.DS↗
arXiv2016

Distinct distances between a collinear set and an arbitrary set of points

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Ω\left(\min\left\{ n^{2/3}m^{2/3},\; \frac{n^{10/11}m^{4/11}}{\log^{2/11}m},\; n^2,\; m^2\right\}\right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.

math.CO↗
arXiv2016

The Elekes-Szabó Theorem in four dimensions

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times C\times D$, unless $F$ has a special group-related form. A similar statement holds for $A,B,C,D$ of unequal sizes. This is a four-dimensional extension of our recent improved analysis of the original Elekes-Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

math.CO↗
arXiv2016

Depth contours in arrangements of halfplanes

Let $H$ be a set of $n$ halfplanes in $\mathbb{R}^2$ in general position, and let $k<n$ be a given parameter. We show that the number of vertices of the arrangement of $H$ that lie at depth exactly $k$ (i.e., that are contained in the interiors of exactly $k$ halfplanes of $H$) is $O(nk^{1/3} + n^{2/3}k^{4/3})$. The bound is tight when $k=Θ(n)$. This generalizes the study of Dey [Dey98], concerning the complexity of a single level in an arrangement of lines, and coincides with it for $k=O(n^{1/3})$.

cs.CG↗
arXiv2016

Incidences between points and lines on two- and three-dimensional varieties

Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in $\mathbb R^4$, such that the points of $P$ lie on an algebraic three-dimensional surface of degree $D$ that does not contain hyperplane or quadric components, and no 2-flat contains more than $s$ lines of $L$. We show that the number of incidences between $P$ and $L$ is $$ I(P,L) = O\left(m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + nD + m\right) , $$ for some absolute constant of proportionality. This significantly improves the bound of the authors, for arbitrary sets of points and lines in $\mathbb R^4$, when $D$ is not too large. The same bound holds when the three-dimensional surface is embedded in any higher dimensional space. For the proof of this bound, we revisit certain parts of [Sharir-Solomon16], combined with the following new 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 two-dimensional algebraic surface of degree $D$ (assumed to be $\ll n^{1/2}$) that does not contain any 2-flat, so that no 2-flat contains more than $s$ lines of $L$ (here we require that the lines of $L$ also be contained in the surface). Then the number of incidences between $P$ and $L$ is $$ I(P,L) = O\left(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + m + n\right). $$ When $d=3$, this improves the bound of Guth and Katz for this special case, when $D \ll n^{1/2}$. Moreover, the bound does not involve the term $O(nD)$, that arises in most standard approaches, and its removal is a significant aspect of our result. Finally, we also obtain (slightly weaker) variants of both results over the complex field. For two-dimensional varieties, the bound is as in the real case, with an added term of $O(D^3)$. For three-dimensional varieties, the bound is as in the real case, with an added term of $O(D^6)$.

math.CO↗
arXiv2016

The Discrete and Semi-continuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection Techniques

The \emph{Fréchet distance} is a well studied similarity measures between curves. The \emph{discrete Fréchet distance} is an analogous similarity measure, defined for a sequence $A$ of $m$ points and a sequence $B$ of $n$ points, where the points are usually sampled from input curves. In this paper we consider a variant, called the \emph{discrete Fréchet distance with shortcuts}, which captures the similarity between (sampled) curves in the presence of outliers. For the \emph{two-sided} case, where shortcuts are allowed in both curves, we give an $O((m^{2/3}n^{2/3}+m+n)\log^3 (m+n))$-time algorithm for computing this distance. When shortcuts are allowed only in one noise-containing curve, we give an even faster randomized algorithm that runs in $O((m+n)^{6/5+\varepsilon})$ expected time, for any $\varepsilon>0$. Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points $A$ and $B$ and an interval $I$, we develop an algorithm that decides whether the number of pairs $(x,y)\in A\times B$ whose distance ${\rm dist}(x,y)$ is in $I$, is less than some given threshold $L$. The running time of this algorithm decreases as $L$ increases. In case there are more than $L$ pairs of points whose distance is in $I$, we can get a small sample of pairs that contains a pair at approximate median distance (i.e., we can approximately "bisect" $I$). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and for computing the semi-continuous Fréchet distance with one-sided shortcuts.

cs.CG↗
arXiv2016

On triple intersections of three families of unit circles

Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szabó [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szabó [Combinatorica, 2012] (and by Elekes and Rónyai [J. Combin. Theory Ser. A, 2000]).

math.MG↗
arXiv2016

Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions

Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir (2005), or only cycles in grid-like patterns, by Chazelle et al. (1992)), and the bounds were considerably weaker---much closer to the trivial quadratic bound. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous "purely combinatorial" methods. Our technique can be extended to eliminating all cycles in the depth relation among segments, and among constant-degree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the much more difficult case of pairwise-disjoint triangles. We also discuss several algorithms for constructing a small set of cuts so as to eliminate all depth-relation cycles among the lines (minimizing such a set, for the case of line segments, is known to be NP-complete). The performance of these algorithms improves due to our new bound, but, so far, none of them both (a) produce close to $n^{3/2}$ cuts, and (b) run in time close to $n^{3/2}$, in the worst case. Our results almost completely settle a 35-year-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

cs.CG↗
arXiv2016

Approximating the $k$-Level in Three-Dimensional Plane Arrangements

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\ovebarline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\Arr}{\cal A}$ Let $H$ be a set of $n$ planes in three dimensions, and let $r \leq n$ be a parameter. We give a simple alternative proof of the existence of a $(1/r)$-cutting of the first $n/r$ levels of $\Arr(H)$, which consists of $O(r)$ semi-unbounded vertical triangular prisms. The same construction yields an approximation of the $(n/r)$-level by a terrain consisting of $O(r/\eps^3)$ triangular faces, which lies entirely between the levels $(1\pm\eps)n/r$. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. The proof is constructive, and leads to a simple randomized algorithm, with expected near-linear running time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane, to obtain a similar construction of "layered" $(1/r)$-cutting of the entire arrangement $\Arr(H)$, of optimal size $O(r^3)$. Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan.

cs.CG↗
arXiv2016

Cutting Algebraic Curves into Pseudo-segments and Applications

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then apply a variant of a technique of Aronov and Sharir to eliminate these depth cycles by making a small number of cuts, which corresponds to a small number of cuts to the original planar arrangement of curves. After these cuts have been performed, the resulting curves form a collection of pseudo-segments. Our cutting bound leads to new incidence bounds between points and constant-degree algebraic curves. The conditions for these incidence bounds are slightly stricter than those for the current best-known bound of Pach and Sharir; for our result to hold, the curves must be algebraic and of bounded maximum degree, while Pach and Sharir's bound only imposes weaker, purely topological constraints on the curves. However, when our conditions hold, the new bounds are superior for almost all ranges of parameters. We also obtain new bounds on the complexity of a single level in an arrangement of constant-degree algebraic curves, and a new bound on the complexity of many marked faces in an arrangement of such curves.

math.CO↗
arXiv2016

Distinct and repeated distances on a surface and incidences between points and spheres

In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $Ω\left(n^{7/9}/{\rm polylog} \,n\right)$. This bound is significantly larger than the conjectured bound $Ω(n^{2/3})$ for general point sets in $\mathbb R^3$. We also show that the number of unit distances determined by $n$ points on a surface $V$, as above, is $O(n^{4/3})$, a bound that matches the best known planar bound, and is worst-case tight in 3-space. This is in sharp contrast with the best known general bound $O(n^{3/2})$ for points in three dimensions. To prove these results, we establish an improved upper bound for the number of incidences between a set $P$ of $m$ points and a set $S$ of $n$ spheres, of arbitrary radii, in $\mathbb R^3$, provided that the points lie on an algebraic surface $V$ of constant degree, which does not have linear or spherical components. Specifically, the bound is $$ O\left( m^{2/3}n^{2/3} + m^{1/2}n^{7/8}\log^β(m^4/n) + m + n + \sum_{c} |P_{c}|\cdot |S_{c}| \right) , $$ where the constant of proportionality and the constant exponent $β$ depend on the degree of $V$, and where the sum ranges over all circles $c$ that are fully contained in $V$, so that, for each such $c$, $P_c = P\cap c$ and $S_c$ is the set of the spheres of $S$ that contain $c$. In addition, $\sum_{c} |P_{c}| = O(m)$ and $\sum_{c} |S_{c}| = O(n)$. This bound too improves upon earlier known bounds. These have been obtained for arbitrary point sets but only under severe restrictions about the spheres, which are dropped in our result. Another interesting application of our result is an incidence bound for arbitrary points and spheres in 3-space, where we improve and generalize the previous work of Apfelbaum and Sharir[AS].

math.CO↗