ź-nets and simplex range queries
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .
These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.
Citations
More filters
Posted Content•
TL;DR: For any fixed dimension d, a random hyperplane search tree with height at most (1 + O(1/sqrt(d)) log 2 n and average element depth at most 2 n with high probability as n \rightarrow \infty was shown in this article.
Abstract: Given a set S of n \geq d points in general position in R^d, a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull A random hyperplane search tree is a binary space partition tree obtained by recursive application of random hyperplane splits We investigate the structural distributions of such random trees with a particular focus on the growth with d A blessing of dimensionality arises--as d increases, random hyperplane splits more closely resemble perfectly balanced splits; in turn, random hyperplane search trees more closely resemble perfectly balanced binary search trees
We prove that, for any fixed dimension d, a random hyperplane search tree storing n points has height at most (1 + O(1/sqrt(d))) log_2 n and average element depth at most (1 + O(1/d)) log_2 n with high probability as n \rightarrow \infty Further, we show that these bounds are asymptotically optimal with respect to d
Posted Content•
TL;DR: An upper bound on the trace function of a hypergraph H is derived and its applications are demonstrated, including a new upper bound for the VC dimension of H that can be used to compute $vc(H)$ in polynomial time provided that $H$ has bounded degeneracy.
Abstract: An upper bound on the trace function of a hypergraph $H$ is derived and its applications are demonstrated. For instance, a new upper bound for the VC dimension of $H$, or $vc(H)$, follows as a consequence and can be used to compute $vc(H)$ in polynomial time provided that $H$ has bounded degeneracy. This was not previously known. Particularly, when $H$ is a hypergraph arising from closed neighborhoods of a graph, this approach asymptotically improves the time complexity of the previous result for computing $vc(H)$. Another consequence is a general lower bound on the {\it distinguishing transversal number } of $H$ that gives rise to applications in domination theory of graphs. To effectively apply the methods developed here, one needs to have good estimations of degeneracy, and its variation or reduced degeneracy which is introduced here.
Posted Content•
TL;DR: In this article, the best known visibility bound for dense and optical forests is obtained by constructing a deterministic digital sequence satisfying strong dispersion properties. But this is not the case for planar dense forests.
Abstract: A 1965 problem due to Danzer asks whether there exists a set in Euclidean space with finite density intersecting any convex body of volume one. A recent approach to this problem is concerned with the construction of dense forests and is obtained by a suitable weakening of the volume constraint. A dense forest is a discrete point set of finite density getting uniformly close to long enough line segments. The distribution of points in a dense forest is then quantified in terms of a visibility function. Another way to weaken the assumptions in Danzer's problem is by relaxing the density constraint. In this respect, a new concept is introduced in this paper, namely that of an optical forest. An optical forest in $\mathbb{R}^{d}$ is a point set with optimal visibility but not necessarily with finite density. In the literature, the best constructions of Danzer sets and dense forests lack effectivity. The goal of this paper is to provide constructions of dense and optical forests which yield the best known results in any dimension $d \ge 2$ both in terms of visibility and density bounds and effectiveness. Namely, there are three main results in this work: (1) the construction of a dense forest with the best known visibility bound which, furthermore, enjoys the property of being deterministic; (2) the deterministic construction of an optical forest with a density failing to be finite only up to a logarithm and (3) the construction of a planar Peres-type forest (that is, a dense forest obtained from a construction due to Peres) with the best known visibility bound. This is achieved by constructing a deterministic digital sequence satisfying strong dispersion properties.
Posted Content•
TL;DR: A more general bound sensitive to the content of $X is shown, which is the first formal justification on why the term $1/\rho$ is not compulsory for "realistic" inputs and constrain $\mathcal{R}$ to be the set of halfspaces in $\mathbb{R]^d$ for a constant $d$.
Abstract: A family $\mathcal{R}$ of ranges and a set $X$ of points together define a range space $(X, \mathcal{R}|_X)$, where $\mathcal{R}|_X = \{X \cap h \mid h \in \mathcal{R}\}$. We want to find a structure to estimate the quantity $|X \cap h|/|X|$ for any range $h \in \mathcal{R}$ with the $(\rho, \epsilon)$-guarantee: (i) if $|X \cap h|/|X| > \rho$, the estimate must have a relative error $\epsilon$; (ii) otherwise, the estimate must have an absolute error $\rho \epsilon$. The objective is to minimize the size of the structure. Currently, the dominant solution is to compute a relative $(\rho, \epsilon)$-approximation, which is a subset of $X$ with $\tilde{O}(\lambda/(\rho \epsilon^2))$ points, where $\lambda$ is the VC-dimension of $(X, \mathcal{R}|_X)$, and $\tilde{O}$ hides polylog factors.
This paper shows a more general bound sensitive to the content of $X$. We give a structure that stores $O(\log (1/\rho))$ integers plus $\tilde{O}(\theta \cdot (\lambda/\epsilon^2))$ points of $X$, where $\theta$ - called the disagreement coefficient - measures how much the ranges differ from each other in their intersections with $X$. The value of $\theta$ is between 1 and $1/\rho$, such that our space bound is never worse than that of relative $(\rho, \epsilon)$-approximations, but we improve the latter's $1/\rho$ term whenever $\theta = o(\frac{1}{\rho \log (1/\rho)})$. We also prove that, in the worst case, summaries with the $(\rho, 1/2)$-guarantee must consume $\Omega(\theta)$ words even for $d = 2$ and $\lambda \le 3$.
We then constrain $\mathcal{R}$ to be the set of halfspaces in $\mathbb{R}^d$ for a constant $d$, and prove the existence of structures with $o(1/(\rho \epsilon^2))$ size offering $(\rho,\epsilon)$-guarantees, when $X$ is generated from various stochastic distributions. This is the first formal justification on why the term $1/\rho$ is not compulsory for "realistic" inputs.
Posted Content•
TL;DR: In this paper, the authors considered the problem of finding an approximate solution to the maximum discrepancy problem in the geometric range space, where the set of ranges defined by the disjoint union of a red and blue set can be represented as a set of halfspaces.
Abstract: Consider the geometric range space $(X, \mathcal{H}_d)$ where $X \subset \mathbb{R}^d$ and $\mathcal{H}_d$ is the set of ranges defined by $d$-dimensional halfspaces. In this setting we consider that $X$ is the disjoint union of a red and blue set. For each halfspace $h \in \mathcal{H}_d$ define a function $\Phi(h)$ that measures the "difference" between the fraction of red and fraction of blue points which fall in the range $h$. In this context the maximum discrepancy problem is to find the $h^* = \arg \max_{h \in (X, \mathcal{H}_d)} \Phi(h)$. We aim to instead find an $\hat{h}$ such that $\Phi(h^*) - \Phi(\hat{h}) \le \varepsilon$. This is the central problem in linear classification for machine learning, in spatial scan statistics for spatial anomaly detection, and shows up in many other areas. We provide a solution for this problem in $O(|X| + (1/\varepsilon^d) \log^4 (1/\varepsilon))$ time, which improves polynomially over the previous best solutions. For $d=2$ we show that this is nearly tight through conditional lower bounds. For different classes of $\Phi$ we can either provide a $\Omega(|X|^{3/2 - o(1)})$ time lower bound for the exact solution with a reduction to APSP, or an $\Omega(|X| + 1/\varepsilon^{2-o(1)})$ lower bound for the approximate solution with a reduction to 3SUM.
A key technical result is a $\varepsilon$-approximate halfspace range counting data structure of size $O(1/\varepsilon^d)$ with $O(\log (1/\varepsilon))$ query time, which we can build in $O(|X| + (1/\varepsilon^d) \log^4 (1/\varepsilon))$ time.
References
More filters
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
"ź-nets and simplex range queries" refers background or methods or result in this paper
...The drawback is that the constants, if deri~,ed from the results in [ 17 ], can be quite large....
[...]
...More generally, we characterize the classes of ranges for which there exists a function f(E) for e S0 such that any finite point set A has an e-net of size f(e), independently of the size of A. These are precisely the classes of ranges with finite Vapnik-Chervonenkis dimension, known as Vapnik-Chervonenkis classes [ 17 ], [9], [19], [1]....
[...]
...The key concepts and proof techniques of this section are based on the pioneering work of Vapnik and Chervonenkis [ 17 ]....
[...]
...Example 5. Let A be a set of n points in E 2. Since the dimension of (E 2, H~-) is 2, the results in [ 17, Theorem 2 ] show that there exists a 0.01-approximation V of A for positive half-planes (and thus for all half-planes) with I VI = 2,525,039....
[...]
...Using the related notion of an e-approxirnation (directly from [ 17 ]), we also point out trivial data structures of constant size that give approximate solutions to the counting problem for halfspaces in constant time (compare [13])....
[...]
Book•
01 Jan 1987TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.
2,284 citations
"ź-nets and simplex range queries" refers background in this paper
...We conclude this section by examining the relationship between the notion of an e-net and the established notion of a centerpoint [21], [11] in combinatorial geometry....
[...]
..., [11] for a general treatment of arrangements....
[...]
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.
1,029 citations
"ź-nets and simplex range queries" refers background in this paper
...Now the assertion can be seen as the dual formulation of Caratheodry's theorem (see [ 15 ], Theorem 2.3.5), which states that if a point x is in the convex hull of a set A in E d, then there exists a subset A' of A such that JA'I -< d + 1 and x is in the convex hull of A'. []...
[...]
TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
Abstract: Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution $P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the empirical measure and let $
u_n := n^\frac{1}{2}(P_n - P)$. Given a class $\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of $
u_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain Gaussian process indexed by $\mathscr{C}$. If convergence holds with respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a suitable (usually nonseparable) function space, we call $\mathscr{C}$ a Donsker class. For measurability, $X$ may be a complete separable metric space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$ does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If $\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then $\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.
555 citations
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.
Abstract: This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.
286 citations
"ź-nets and simplex range queries" refers methods in this paper
...It should be noted that better bounds are possible for reporting in two dimensions (specifically O(log n + t) time, where t is the number of points reported [3]), but these techniques only work for half-planes....
[...]