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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Posted Content•
TL;DR: The notion of relative $(p,\eps)$-approximations is re-examine, recently introduced in [CKMS06], and upper bounds on their size are established, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies.
Abstract: We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies [Pol86, Hau92, Tal94]. We also survey the different notions of sampling, used in computational geometry, learning, and other areas, and show how they relate to each other. We then give constructions of smaller-size relative $(p,\eps)$-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure--spanning trees with small relative crossing number, which we believe to be of independent interest. Relative $(p,\eps)$-approximations arise in several geometric problems, such as approximate range counting, and we apply our new structures to obtain efficient solutions for approximate range counting in three dimensions. We also present a simple solution for the planar case.
9 citations
01 Jan 1994
TL;DR: This work examines the complexity of testing different program constructs by defining a measure of testing complexity known as VCP-dimension, which is similar to the Vapnik?Chervonenkis dimension, and applying it to classes of programs, where all programs in a class share the same syntactic structure.
Abstract: We examine the complexity of testing different program constructs. We do this by defining a measure of testing complexity known as VCP-dimension, which is similar to the Vapnik-Chervonenkis dimension, and applying it to classes of programs, where all programs in a class share the same syntactic structure. VCP-dimension gives bounds on the number of test points needed to determine that a program is approximately correct, so by studying it for a class of programs we gain insight into the difficulty of testing the program construct represented by the class. We investigate the VCP-dimension of straight line code, if-then-else statements, and for loops. We also compare the VCP-dimension of nested and sequential if-then-else statements as well as that of two types of for loops with embedded if-then-else statements. Finally, we perform an empirical study to estimate the expected complexity of straight line code.
9 citations
17 Nov 2008
TL;DR: This work studies the decision problem of disjunctive linear arithmetic over the reals from the perspective of computational geometry and shows a branching heuristic that is based on approximating T-implications, based on a geometric analysis.
Abstract: We study the decision problem of disjunctive linear arithmetic over the reals from the perspective of computational geometry. We show that traversing the linear arrangement induced by the formula's predicates, rather than the DPLL(T) method of traversing the Boolean space, may have an advantage when the number of variables is smaller than the number of predicates (as it is indeed the case in the standard SMT-Lib benchmarks). We then continue by showing a branching heuristic that is based on approximating T-implications, based on a geometric analysis. We achieve modest improvement in run time comparing to the commonly used heuristic used by competitive solvers.
9 citations
Cites background from "ź-nets and simplex range queries"
...We can learn about the potential of theory propagation in the case of LRA, that is, the average number of implications given a partial assignment, from a theoretical result in computational geometry by Haussler and Welzl [ 13 ]....
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...In a series of papers, Haussler and Welzl [ 13 ] and Clarkson [5], suggested that for a n × d linear system, if we randomly select r constraints such that d ≤ r<n , and build a linear arrangement, then with high probability ‐ a probability of (1−1/r c ), where c is a constant ‐ the interior of any cell in...
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TL;DR: This work solves some problems related toray shooting in the plane, such as finding the first object hit by a query ray or counting the number of objects intersected by the query line with an algorithm for finding thefirst hit when the objects are lines.
Abstract: We solve some problems related toray shooting in the plane, such as finding the first object hit by a query ray or counting the number of objects intersected by the query line. Our main results are an algorithm for finding the first hit when the objects are lines, and an algorithm for the case when the objects are segments. If the segments form simple polygons, this information can be used for reducing the complexity of the algorithms. The algorithms are efficient in space and in query time. Moreover, they are simple and therefore of practical use.
8 citations
07 Oct 2018
TL;DR: In this article, a polynomial-time algorithm was proposed for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number.
Abstract: We propose a polynomial-time algorithm which takes as input a finite set of points of R^3 and computes, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii, in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. Recently, it was shown that the disjoint union of two odd cycles is never the complement of a disk graph [Bonnet, Giannopoulos, Kim, Rzazewski, Sikora; SoCG '18]. This enabled the authors to derive a QPTAS and a subexponential algorithm for Max Clique on disk graphs. In this paper, we improve the approximability to a randomized EPTAS (and a deterministic PTAS). More precisely, we obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. We then address the question of computing Max Clique for disks in higher dimensions. We show that intersection graphs of unit balls, like disk graphs, do not admit the complement of two odd cycles as an induced subgraph. This, in combination with the first result, straightforwardly yields a randomized EPTAS for Max Clique on unit ball graphs. In stark contrast, we show that on ball graphs and unit 4-dimensional disk graphs, Max Clique is NP-hard and does not admit an approximation scheme even in subexponential-time, unless the Exponential Time Hypothesis fails.
8 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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