ź-nets and simplex range queries
Citations
11 citations
Cites background from "ź-nets and simplex range queries"
...Weak ε-nets with respect to convex sets as defined in the abstract were introduced by Haussler and Welzl [10] and later found many applications in discrete geometry, most notably in the spectacular proof of the Hadwiger-Debrunner (p, q)-conjecture by Alon and Kleitman [3]....
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11 citations
Cites background from "ź-nets and simplex range queries"
...Haussler and Welzl have shown [13] that if the V-C dimension of H(X,Y ) is some integer d, then for every > 0 there is an -net of size at most cd −1 ln −1, where c is a small constant....
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10 citations
10 citations
Cites background from "ź-nets and simplex range queries"
...One may phrase the definition of a (t, g)-trimming of an unweighted n-vertex graph G = (V ,E) in the language of ε-nets [9]....
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10 citations
Cites background or methods from "ź-nets and simplex range queries"
...In [8] it is shown that every family of bounded VC-dimension has a (p, q) theorem....
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...This result in [8] in fact uses the Alon-Kleitman method in [3] together with an established fractional Helly number of families with bounded (dual) VC-dimension, and the existence of a bounded -net for families of bounded VC-dimension [5]....
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...In the proof of Theorem 1 we will follow the footsteps of Matoušek in [8] where the fractional Helly number of FP is bounded in terms of the dual VC-dimension of F and more specifically, in terms of the dual shatter function of F ....
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...In [8] Matoušek shows that families of sets with bounded dual VC-dimension have a (bounded) fractional Helly number (see definition below)....
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...This includes in particular the theory of -nets and weak -nets [5]....
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References
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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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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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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555 citations
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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