# 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 Ã‚-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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### Cites background from "ź-nets and simplex range queries"

...5 are taken from Preparata, Hong (1977); another presentation of the same algorithms can be found in Preparata, Shamos (1985). The variation of their two-dimensional algorithm described in Section 8....

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...5 are taken from Preparata, Hong (1977); another presentation of the same algorithms can be found in Preparata, Shamos (1985). The variation of their two-dimensional algorithm described in Section 8.3.2 guarantees running time O( n) if the points are presorted; this feature is not shared by the original design. The presentation of the three-dimensional algorithm in Section 8.5 is the first complete description of the algorithm given in the literature. Until now, it has not been overlooked that a single vertex of a recursively constructed convex polytope can be encountered more than once when it is merged with another disjoint convex polytope. Due to the close relationship between convex hulls in three dimensions and Voronoi diagrams in two dimensions, it is not surprising that the corresponding phenomenon was overlooked in the analysis of divide-and-conquer algorithms for constructing Voronoi diagrams (see Shamos (1978), Lee (1978), Preparata, Shamos (1985), and others)....

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1,835 citations

### Cites background or methods from "ź-nets and simplex range queries"

...Using Proposition A2.5, they generalize Lemmas 3.4 and 3.5 of [ 29 ] to arbitrary probability distributions....

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...introduced in [ 29 ]* to arbitrary probability distributions on E”. For a fixed distribution, an E-transversal for R is a finite set of points N G E” such that every region in R of probability at least E contains at least one point in N. Se:ction A2 uses the notion of an c-transversal to provide the primary machinery for Theorem 2.1, following [29] and [62]....

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...We sketch the proof for completeness, using the notation from [ 29 ]....

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...A. BLUMER ET AL. function II,(m) (see [ 29 ])....

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...Our characterization of learnability uses a simple combinatorial parameter called the Vapnik-Chervonenkis (VC) dimension of the class C of concepts [ 29 ].’ We show that there is a learning function satisfying (3) if and only if the VC dimension of C is finite....

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##### References

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### "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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2,269 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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937 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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540 citations

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### "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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