Epsilon-nets and simplex range queries

Haussler, David; Welzl, Emo

doi:10.1145/10515.10522

Cited by 217 publications

(316 citation statements)

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“…In general, range spaces involving semi-algebraic ranges of constant description complexity, i.e., semi-algebraic sets defined as a Boolean combination of a constant number of polynomial equations and inequalities of constant maximum degree, have finite VC-dimension. Halfspaces, balls, ellipsoids, simplices, and boxes are examples of ranges of this kind; see [12,22,34,37] for definitions and more details.…”

Section: Introductionmentioning

confidence: 99%

Relative (p,ε)-Approximations in Geometry

Har-Peled

Sharir

2010

Discrete Comput Geom

141

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We re-examine the notion of relative (p, ε)-approximations, recently introduced in Cohen et al. (Manuscript, 2006), and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in Li et al. (J. Comput. Syst. Sci. 62:516-527, 2001) and in several earlier studies (Pollard in Manuscript, 1986; Haussler in Inf. Comput. 100:78-150, 1992; Talagrand in Ann. Probab. 22:28-76, 1994). 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, ε)-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, ε)-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.

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Section: Introductionmentioning

confidence: 99%

Relative (p,ε)-Approximations in Geometry

Har-Peled

Sharir

2010

Discrete Comput Geom

141

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“…The restriction of Γ to a set Y ⊆ X, denoted by Γ| Y , is the set {Y ∩ R | R ∈ Γ}. An important measure for the complexity of a range space is its VC-dimension: A range space R has VC-dimension z if there exists a subset Y ⊆ X of maximal cardinality z such that Γ| Y equals the power-set of Y [10]. Though range spaces are formulated on a set theoretic level, in this paper we only need the special case for which X = R d .…”

Section: A Unified Framework For the Gnn Problemmentioning

confidence: 99%

“…For all R ∈ Γ, if |R ∩ P | ≥ ε |P | then R ∩ N = ∅. This concept was first introduced to computational geometry by Haussler and Welzl [10], but had also significant impact on other fields of computer science. To simplify notation an ε-net of P is denoted by N ε (P ).…”

Section: A Unified Framework For the Gnn Problemmentioning

confidence: 99%

“…All line numbers refer to Algorithm 1. At the highest depth of recursion the algorithm works on (P k , ∅) and searches P k for a closest point to Q (line 6, line [8][9][10][11][12]. Thus, we get T (|P k |) = T D |P k |.…”

Section: The Frameworkmentioning

confidence: 99%

“…First, by T (|P i+1 |) which is the running time of the recursive call on P i+1 (line 3). Then, by T R (|P i | , |R|) which is the running time of the range searching algorithm A R on P i , and finally, by the time needed to search the resulting set R for the closest point to Q which is bounded by T D |R| (line [8][9][10][11][12]. From this we get…”

Section: The Frameworkmentioning

confidence: 99%

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A Simple Framework for the Generalized Nearest Neighbor Problem

Hrúz

Schöngens

2012

Algorithm Theory – SWAT 2012

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The problem of finding a nearest neighbor from a set of points in R d to a complex query object has attracted considerable attention due to various applications in computational geometry, bio-informatics, information retrieval, etc. We propose a generic method that solves the problem for various classes of query objects and distance functions in a unified way. Moreover, for linear space requirements the method simplifies the known approach based on ray-shooting in the lower envelope of an arrangement.

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Core‐sets: An updated survey

Feldman

2019

WIREs Data Min & Knowl

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In optimization or machine learning problems we are given a set of items, usually points in some metric space, and the goal is to minimize or maximize an objective function over some space of candidate solutions. For example, in clustering problems, the input is a set of points in some metric space, and a common goal is to compute a set of centers in some other space (points, lines) that will minimize the sum of distances to these points. In database queries, we may need to compute such a some for a specific query set of k centers. However, traditional algorithms cannot handle modern systems that require parallel real‐time computations of infinite distributed streams from sensors such as GPS, audio or video that arrive to a cloud, or networks of weaker devices such as smartphones or robots. Core‐set is a “small data” summarization of the input “big data,” where every possible query has approximately the same answer on both data sets. Generic techniques enable efficient coreset maintenance of streaming, distributed and dynamic data. Traditional algorithms can then be applied on these coresets to maintain the approximated optimal solutions. The challenge is to design coresets with provable tradeoff between their size and approximation error. This survey summarizes such constructions in a retrospective way, that aims to unified and simplify the state‐of‐the‐art. This article is categorized under Algorithmic Development > Structure Discovery Fundamental Concepts of Data and Knowledge > Big Data Mining Technologies > Machine Learning Algorithmic Development > Scalable Statistical Methods

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Epsilon-nets and simplex range queries

Cited by 217 publications

References 10 publications

Relative (p,ε)-Approximations in Geometry

Relative (p,ε)-Approximations in Geometry

A Simple Framework for the Generalized Nearest Neighbor Problem

Core‐sets: An updated survey

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