ź-nets and simplex range queries

doi:10.1007/BF02187876

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ź-nets and simplex range queries

Journal Article•DOI•

ź-nets and simplex range queries

David Haussler¹, Emo Welzl²•Institutions (2)

University of California, Santa Cruz¹, University of Graz²

01 Dec 1987-Discrete and Computational Geometry (Springer New York)-Vol. 2, Iss: 1, pp 127-151

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 Ã‚-net with any desired probability are given.

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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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Geometric Arrangements: Substructures and Algorithms

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Beverly Sackler

01 Jan 2007

3 citations

Journal Article•DOI•

Capacitated discrete unit disk cover

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Pawan Kumar Mishra¹, Sangram K. Jena¹, Gautam K. Das¹, Seela Veerabhadreswara Rao¹•Institutions (1)

Indian Institute of Technology Guwahati¹

15 Oct 2020-Discrete Applied Mathematics

TL;DR: The geometric version of the capacitated covering problem, namely, capacitated discrete unit disk cover problem, is considered and it is proved that the ( α, P, Q ) -covering problem is NP-complete for α ≥ 3 and the problem admits a PTAS.

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3 citations

Posted Content•

Higher Order Function Networks for View Planning and Multi-View Reconstruction

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Selim Engin¹, Eric Mitchell², Daewon Lee², Volkan Isler², Daniel D. Lee² - Show less +1 more•Institutions (2)

University of Minnesota¹, Samsung²

04 Oct 2019-arXiv: Robotics

TL;DR: This work extends a recent method which uses Higher Order Functions (HOF) to represent the shape of the object and presents a new generalization of this method to incorporate multiple images as input and establish a connection between visibility and reconstruction quality.

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Abstract: We consider the problem of planning views for a robot to acquire images of an object for visual inspection and reconstruction. In contrast to offline methods which require a 3D model of the object as input or online methods which rely on only local measurements, our method uses a neural network which encodes shape information for a large number of objects. We build on recent deep learning methods capable of generating a complete 3D reconstruction of an object from a single image. Specifically, in this work, we extend a recent method which uses Higher Order Functions (HOF) to represent the shape of the object. We present a new generalization of this method to incorporate multiple images as input and establish a connection between visibility and reconstruction quality. This relationship forms the foundation of our view planning method where we compute viewpoints to visually cover the output of the multi-view HOF network with as few images as possible. Experiments indicate that our method provides a good compromise between online and offline methods: Similar to online methods, our method does not require the true object model as input. In terms of number of views, it is much more efficient. In most cases, its performance is comparable to the optimal offline case even on object classes the network has not been trained on.

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3 citations

Cites background from "ź-nets and simplex range queries"

...If there is an ε such that V (x) contains a disk of area ε for all x ∈ X , then from [23] we know that n = O(Aε log A ε ) samples suffice to cover all the points, where A is the surface area of the viewing space....
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...A principled way of determining n, the number of sampled viewpoints, is through the notion of ε-nets [23], [24]....
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Journal Article•DOI•

Algorithms for finding attribute value group for binary segmentation of categorical databases

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Yasuhiko Morimoto¹, Takeshi Fukuda¹, Takeshi Tokuyama•Institutions (1)

IBM¹

01 Nov 2002-IEEE Transactions on Knowledge and Data Engineering

TL;DR: Though the problem is intractable for general objective functions, there are feasible algorithms for finding high quality binary segmentations when the objective function is convex, and it is proved that the typical criteria mentioned above are all convex.

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Abstract: We consider the problem of finding a set of attribute values that give a high quality binary segmentation of a database. The quality of a segmentation is defined by an objective function suitable for the user's objective, such as "mean squared error," "mutual information," or "/spl chi//sup 2/" each of which is defined in terms of the distribution of a given target attribute. Our goal is to find value groups on a given conditional domain that split databases into two segments, optimizing the value of an objective function. Though the problem is intractable for general objective functions, there are feasible algorithms for finding high quality binary segmentations when the objective function is convex, and we prove that the typical criteria mentioned above are all convex. We propose two practical algorithms, based on computational geometry techniques, which find a much better value group than conventional heuristics.

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3 citations

Cites background from "ź-nets and simplex range queries"

...points, roughly speaking, because the Vapnik-Chervonenkis dimension is k and, hence, it is known [ 15 ], [2] that a random sample of size...
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Journal Article•DOI•

Building an Optimal Point-Location Structure in $$O( sort (n))$$ O ( s o r t ( n ) ) I/Os

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Xiaocheng Hu¹, Cheng Sheng¹, Yufei Tao¹•Institutions (1)

The Chinese University of Hong Kong¹

01 May 2019-Algorithmica

TL;DR: This work presents the first algorithm that solves the problem of constructing an external memory data structure on a planar subdivision formed by n segments to answer point location queries optimally in O(logBn) I/Os deterministically.

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Abstract: We revisit the problem of constructing an external memory data structure on a planar subdivision formed by n segments to answer point location queries optimally in $$O(\log _B n)$$ I/Os. The objective is to achieve the I/O cost of $$ sort (n) = O(\frac{n}{B} \log _{M/B} \frac{n}{B})$$ , where B is the number of words in a disk block, and M being the number of words in memory. The previous algorithms are able to achieve this either in expectation or under the tall cache assumption of $$M \ge B^2$$ . We present the first algorithm that solves the problem deterministically for all values of M and B satisfying $$M \ge 2B$$ .

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3 citations

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References

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Book Chapter•DOI•

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

[...]

Vladimir Vapnik, A. Ya. Chervonenkis

01 Jan 1971-Theory of Probability and Its Applications

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.

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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).

...read moreread less

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....
[...]
...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•

Algorithms in Combinatorial Geometry

[...]

Herbert Edelsbrunner¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

01 Jan 1987

TL;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.

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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.

...read moreread less

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....
[...]

Journal Article•DOI•

On the density of families of sets

[...]

Norbert Sauer¹•Institutions (1)

University of Calgary¹

01 Jul 1972-Journal of Combinatorial Theory, Series A

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.

...read moreread less

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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Journal Article•DOI•

Central Limit Theorems for Empirical Measures

[...]

Richard M. Dudley

01 Dec 1978-Annals of Probability

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.

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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 $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

...read moreread less

555 citations

Journal Article•DOI•

The power of geometric duality

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Bernard Chazelle¹, Leonidas J. Guibas², Der-Tsai Lee³•Institutions (3)

Brown University¹, PARC², Northwestern University³

01 Jun 1985-Bit Numerical Mathematics

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.

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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.

...read moreread less

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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