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.
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04 May 1997
TL;DR: A memorial to Paul Erd6s, whose oeuw-e encompasses a multitude of areas of mathematics, including combinatorics, set theory, number theory, classical analysis, discrete geometry, probability theory, and more.
Abstract: Paul Erd6s’s oeuw-e encompasses a multitude of areas of mathematics, including combinatorics, set theory, number theory, classical analysis, discrete geometry, probability theory, and more. The theory of computing is conspicuously missing from this list. It is a field in which Erd6s never took any interest. How, then, did Erd6s become a household name in the theoretical computer science community? We address this question in this memorial.
2 citations
Journal Article•
TL;DR: An algorithm is presented that, for any @e>0, finds an overlap at least the optimum minus @e and reports the translation realizing it and computes the maximum overlap of the perturbed polytopes.
Abstract: We study the problem of maximizing the overlap of two convex polytopes under translation in R^d for some constant d>=3 Let n be the number of bounding hyperplanes of the polytopes We present an algorithm that, for any @e>0, finds an overlap at least the optimum minus @e and reports the translation realizing it The running time is O(n^@?^d^/^2^@?^+^1log^dn) with probability at least 1-n^-^O^(^1^), which can be improved to O(nlog^3^^5n) in R^3 The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes The perturbation causes an additive error @e, which can be made arbitrarily small by decreasing the perturbation magnitude Our algorithm in fact computes the maximum overlap of the perturbed polytopes The running time bounds, the probability bound, and the big-O constants in these bounds are independent of @e
2 citations
TL;DR: The VC-dimension of the set system on the vertex set of some graph which is induced by the family of its k -connected subgraphs is studied and it is shown that computing the VC- dimension is NP -complete and that it remainsNP -complete for split graphs and for some subclasses of planar bipartite graphs.
2 citations
Cites background from "ź-nets and simplex range queries"
...Since a graph of order n has n closed neighbourhoods, then its VC-dimension is at most blog2 nc [2]....
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...This study was first initiated in a seminal paper by Haussler and Welzl [2]....
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01 Sep 2021
TL;DR: This work provides the first strongly polynomial (in both $n$ and $d$) approximation algorithm for finding a Tverberg point, and provides several new approximation algorithms for this problem, which improve either the running time or quality of approximation.
Abstract: Tverberg’s theorem states that a set of n points in ℝ^d can be partitioned into ⌈n/(d+1)⌉ sets whose convex hulls all intersect. A point in the intersection (aka Tverberg point) is a centerpoint, or high-dimensional median, of the input point set. While randomized algorithms exist to find centerpoints with some failure probability, a partition for a Tverberg point provides a certificate of its correctness.
Unfortunately, known algorithms for computing exact Tverberg points take n^{O(d²)} time. We provide several new approximation algorithms for this problem, which improve running time or approximation quality over previous work. In particular, we provide the first strongly polynomial (in both n and d) approximation algorithm for finding a Tverberg point.
2 citations
TL;DR: A general method is presented which allows using a (p, 2)-theorem as a bootstrapping to obtain a tight ( p, q)-theorems, for classes with Helly number 2, even without assuming that the sets in the class are convex or compact.
Abstract: A family $$\mathcal {F}$$ of sets is said to satisfy the (p, q)-property if among any p sets of $$\mathcal {F}$$ some q have a non-empty intersection. The celebrated (p, q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in $$\mathbb {R}^d$$ that satisfies the (p, q)-property for some $$q \ge d+1$$, can be pierced by a fixed number (independent of the size of the family) $$f_d(p,q)$$ of points. The minimum such piercing number is denoted by $$\mathsf {HD} _d(p,q)$$. Already in 1957, Hadwiger and Debrunner showed that whenever $$q>\frac{d-1}{d}\,p+1$$ the piercing number is $$\mathsf {HD} _d(p,q)=p-q+1$$; no tight bounds on $$\mathsf {HD} _d(p,q)$$ were found ever since. While for an arbitrary family of compact convex sets in $$\mathbb {R}^d$$, $$d \ge 2$$, a (p, 2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific classes. The best-studied among them is the class of axis-parallel boxes in $$\mathbb {R}^d$$, and specifically, axis-parallel rectangles in the plane. Wegner (Israel J Math 3:187–198, 1965) and (independently) Dol’nikov (Sibirsk Mat Ž 13(6):1272–1283, 1972) used a (p, 2)-theorem for axis-parallel rectangles to show that $$\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$$ holds for all $$q \ge \sqrt{2p}$$. These are the only values of q for which $$\mathsf {HD} _\mathrm{{rect}}(p,q)$$ is known exactly. In this paper we present a general method which allows using a (p, 2)-theorem as a bootstrapping to obtain a tight (p, q)-theorem, for classes with Helly number 2, even without assuming that the sets in the class are convex or compact. To demonstrate the strength of this method, we show that $$\mathsf {HD} _{d\text {-box}}(p,q)=p-q+1$$ holds for all $$q > c' \log ^{d-1} p$$, and in particular, $$\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1$$ holds for all $$q \ge 7 \log _2 p$$ (compared to $$q \ge \sqrt{2p}$$, obtained by Wegner and Dol’nikov more than 40 years ago). In addition, for several classes, we present improved (p, 2)-theorems, some of which can be used as a bootstrapping to obtain tight (p, q)-theorems. In particular, we show that any class $$\mathcal {G}$$ of compact convex sets in $$\mathbb {R}^d$$ with Helly number 2 admits a (p, 2)-theorem with piercing number $$O(p^{2d-1})$$, and thus, satisfies $$\mathrm {HD}_{\mathcal {G}}(p,q) = p-q+1$$, for a universal constant c.
2 citations
References
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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....
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...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]....
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...The key concepts and proof techniques of this section are based on the pioneering work of Vapnik and Chervonenkis [ 17 ]....
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...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....
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...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])....
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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....
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..., [11] for a general treatment of arrangements....
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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'. []...
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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....
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