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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TL;DR: In this paper, the authors obtained a superlinear lower bound for weak 1/r -net with at least Ω(r logd−1 r) points, where r is the Cartesian product of d suitable fast growing finite sequences.
Abstract: A set N ⊂ ℝ
d
is called a weak
ɛ-net (with respect to convex sets) for a finite X ⊂ ℝ
d
if N intersects every convex set C with |X ∩ C| ≥ ɛ|X|. For every fixed d ≥ 2 and every r ≥ 1 we construct sets X ⊂ ℝ
d
for which every weak 1/r -net has at least Ω(r logd−1
r) points; this is the first superlinear lower bound for weak ɛ-nets in a fixed dimension. The construction is a stretched grid, i.e., the Cartesian product of d suitable fast-growing finite sequences, and convexity in this grid can be analyzed using stair-convexity, a new variant of the usual notion of convexity. We also consider weak ɛ-nets for the diagonal of our stretched grid in ℝ
d
, d ≥ 3, which is an “intrinsically 1-dimensional” point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that the upper bounds by Alon, Kaplan, Nivasch, Sharir and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the known upper bound for the so-called second selection lemma in the plane by a logarithmic factor: We obtain a set T of t triangles with vertices in an n-point set in the plane such that no point is contained in more than O(t
2/(n
3 logn
3/t)) triangles of T.
50 citations
TL;DR: It is proved that, for any constant ɛ>0, the complexity of the vertical decomposition of a set ofn triangles in three-dimensional space isO(n2+ɛ+K), whereK is the complexityof the arrangement of the triangles.
Abstract: We prove that, for any constant ɛ>0, the complexity of the vertical decomposition of a set ofn triangles in three-dimensional space isO(n
2+ɛ
+K), whereK is the complexity of the arrangement of the triangles. For a single cell the complexity of the vertical decomposition is shown to beO(n
2+ɛ
). These bounds are almost tight in the worst case. We also give a deterministic output-sensitive algorithm for computing the vertical decomposition that runs inO(n
2
logn+V logn) time, whereV is the complexity of the decomposition. The algorithm is reasonably simple (in particular, it tries to perform as much of the computation in two-dimensional spaces as possible) and thus is a good candidate for efficient implementations. The algorithm is extended to compute the vertical decomposition of arrangements ofn algebraic surface patches of constant maximum degree in three-dimensional space in timeO(nλ
q
(n) logn +V logn), whereV is the combinatorial complexity of the vertical decomposition, λ
q
(n) is a near-linear function related to Davenport-Schinzel sequences, andq is a constant that depends on the degree of the surface patches and their boundaries. We also present an algorithm with improved running time for the case of triangles which is, however, more complicated than the first algorithm.
50 citations
Cites background or methods from "ź-nets and simplex range queries"
...From e-net theory [ 26 ] it follows that with high probability each simplex in the triangulation will be intersected by O (n log r/r) triangles in T. In other words, the triangulation will be an O (log r/r)cutting for T with high probability....
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...vertical decomposition of a set T of n triangles in 3-space is O (n 2+e + K), where K is the complexity of the arrangement ,A(T) induced by T. Our proof uses an interesting combination of efficient hierarchical cuttings [8], [29], the counting scheme used in hereditary segment trees [11], the Slicing Theorem [4], and random sampling [16], [ 26 ]....
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01 May 1996
TL;DR: Two results from Clarkson’s randomized algorithm for linear programming in a fixed dimension d are derived, a simple general method that reduces the problem of answering linear programming queries to theproblem of answering halfspace range queries and a simpler proof of the following.
Abstract: We derive two results from Clarkson’s randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a randomized data structure with O(n) space and 0(nl-li[d12~ 2°f10g* ‘)) query time for linear programming on n halfspaces (d > 3). The second result is a simpler proof of the following: a sequence of q linear programming queries on n halfspaces can be answered in O (n log q) time, if q s nud for a certain const ant ad > 0. Unlike previous methods, our algorithms do not require parametric searching.
49 citations
TL;DR: This work presents a Garey/Johnson-style list of problems known to be complete for the second and higher levels of the polynomial-time hierarchy (polynomial hierarchy, or PH for short).
Abstract: We present a Garey/Johnson-style list of problems known to be complete for the second and higher levels of the polynomial-time hierarchy (polynomial hierarchy, or PH for short). We also include the best-known hardness of approximation results. The list currently does not contain results on nonclassical logics and databases.
49 citations
Cites background from "ź-nets and simplex range queries"
...Introduced by Kranakis, et al. [25] building on an idea of Haussler and Welzl [ 15 ]....
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01 May 1990
TL;DR: Borders are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three by solving the problem of Haussler and Welzl by showing that if d ≥ 2, then ƒ1 = max(2,⌈1/ε
Abstract: Given any natural number d, 0 e d(e) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension d has an e-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if d ≥ 2, then ƒd(e) > 1/48 d/e log 1/ e which is not far from being optimal, if d is fixed and e → 0. Further, we prove that ƒ1(e) = max(2,⌈1/e⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.
49 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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