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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20 Aug 2012
TL;DR: In this paper, the authors studied the Partial Nearest Neighbor Problem (PNNP) and proposed an exact algorithm with linear space and sub-linear worst-case query time for l 2 and l ∞ -metrics.
Abstract: We study the Partial Nearest Neighbor Problem that consists in preprocessing n points \(\mathcal{D}\) from d-dimensional metric space such that the following query can be answered efficiently: Given a query vector Q ∈ ℝ d and an axes-aligned query subspace represented by S ∈ {0,1} d , report a point \(P \in \mathcal{D}\) with d S (Q,P) ≤ d S (Q,P′) for all \(P' \in \mathcal{D}\), where d S (Q,P) is the distance between Q and P in the subspace S. This problem is related to similarity search between feature vectors w.r.t. a subset of features. Thus, the problem is of great practical importance in bioinformatics, image recognition, etc., however, due to exponentially many subspaces, each changing distances significantly, the problem has a considerable complexity. We present the first exact algorithms for l2- and l ∞ -metrics with linear space and sub-linear worst-case query time. We also give a simple approximation algorithm, and show experimentally that our approach performs well on real world data.
1 citations
Posted Content•
TL;DR: For general point sets in dimension $d\geq 3, it was shown in this article that for any finite point set P in the Euclidean space, and any point set ϵ > 0, there exists a weak ϵ-net of cardinality O(1/2+ ϵ ϵ + ε ≥ 0, where ϵ>0 is an arbitrary small constant.
Abstract: Given a finite point set $P$ in ${\mathbb R}^d$, and $\epsilon>0$ we say that $N\subseteq {\mathbb R}^d$ is a weak $\epsilon$-net if it pierces every convex set $K$ with $|K\cap P|\geq \epsilon |P|$.
Let $d\geq 3$. We show that for any finite point set in ${\mathbb R}^d$, and any $\epsilon>0$, there exist a weak $\epsilon$-net of cardinality $\displaystyle O\left(\frac{1}{\epsilon^{d-1/2+\gamma}}\right)$, where $\gamma>0$ is an arbitrary small constant.
This is the first improvement of the bound of $\displaystyle O^*\left(\frac{1}{\epsilon^d}\right)$ that was obtained in 1994 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension $d\geq 3$.
1 citations
01 Jan 1994
1 citations
Cites background from "ź-nets and simplex range queries"
...of Vapnik and Chervonenkis [69] and Haussler and Welzl [ 38 ], have had a...
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Posted Content•
TL;DR: It is proved that any sufficiently large hypergraph with VC-dimension $d admits an $\epsilon$-$t$-net of size $O(\frac{ (1+\log t)d}{\ep silon} \log \frac{1}{\ epsilon})$.
Abstract: We study a natural generalization of the classical $\epsilon$-net problem (Haussler--Welzl 1987), which we call the "$\epsilon$-$t$-net problem": Given a hypergraph on $n$ vertices and parameters $t$ and $\epsilon\geq \frac t n$, find a minimum-sized family $S$ of $t$-element subsets of vertices such that each hyperedge of size at least $\epsilon n$ contains a set in $S$. When $t=1$, this corresponds to the $\epsilon$-net problem.
We prove that any sufficiently large hypergraph with VC-dimension $d$ admits an $\epsilon$-$t$-net of size $O(\frac{ (1+\log t)d}{\epsilon} \log \frac{1}{\epsilon})$. For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $O(\frac{1}{\epsilon})$-sized $\epsilon$-$t$-nets.
We also present an explicit construction of $\epsilon$-$t$-nets (including $\epsilon$-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of $\epsilon$-nets (i.e., for $t=1$), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.
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Posted Content•
TL;DR: The proof is based on showing a similar result for families $\mathcal{F}$ of sets separable by pseudo-discs in $\mathbb{R}^2$ and it is complemented by showing that analogous result fails to hold for collections of linearly separable sets in 𝕂R^4 and higher dimensional euclidean spaces.
Abstract: Let $\mathcal{F}$ be any collection of linearly separable sets of a set $P$ of $n$ points either in $\mathbb{R}^2$, or in $\mathbb{R}^3$. We show that for every natural number $k$ either one can find $k$ pairwise disjoint sets in $\mathcal{F}$, or there are $O(k)$ points in $P$ that together hit all sets in $\mathcal{F}$. The proof is based on showing a similar result for families $\mathcal{F}$ of sets separable by pseudo-discs in $\mathbb{R}^2$. We complement these statements by showing that analogous result fails to hold for collections of linearly separable sets in $\mathbb{R}^4$ and higher dimensional euclidean spaces.
1 citations
Cites background from "ź-nets and simplex range queries"
...Therefore, each Hi has an -net of size that depends only on (see [17])....
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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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