Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Barone, Sal; Basu, Saugata

doi:10.1007/s00454-011-9391-3

Cited by 44 publications

(101 citation statements)

References 26 publications

(43 reference statements)

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“…In the latter case we apply a generalization of a classical result established independently by Oleinik and Petrovsky [32], Milnor [29], and Thom [51], such that the number of connected components of ℓ\{Q = 0} is at most O(C k 1 ); we give a proof of this fact in Theorem A.2. Recently a more general bound was proved by Barone and Basu [4].…”

Section: L|mentioning

confidence: 96%

See 1 more Smart Citation

An Incidence Theorem in Higher Dimensions

Solymosi

Tao²

2012

Discrete Comput Geom

114

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Section: L|mentioning

confidence: 96%

“…) Recently a more general bound was presented by Barone and Basu [4]. They gave tight estimates on the dependence on the various parameters which are hidden in the big-Oh notation in Theorem A.2.…”

Section: This Implies Thatmentioning

confidence: 98%

An Incidence Theorem in Higher Dimensions

Solymosi

Tao²

2012

Discrete Comput Geom

114

143

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“…The state-of-the-art upper bounds are by (i) Chan [6], who, building on many earlier results, provides a linear-size data structure with O(n log n) expected preprocessing time and O(n 1−1/d ) query time, and (ii) Matoušek [22], who provides a data structure with O(n d ) storage, O((log n) d+1 ) query time, and O(n d (log n) ε ) preprocessing time. 3 A trade-off between space and query time can be obtained by combining these two data structures [22].…”

Section: Introductionmentioning

confidence: 99%

On Range Searching with Semialgebraic Sets II

Agarwal

#x²,

Matouek³

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Let P be a set of n points in R d . We present a linear-size data structure for answering range queries on P with constant-complexity semialgebraic sets as ranges, in time close to O(n 1−1/d ). It essentially matches the performance of similar structures for simplex range searching, and, for d ≥ 5, significantly improves earlier solutions by the first two authors obtained in 1994. This almost settles a long-standing open problem in range searching.The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter r, 1 < r ≤ n, there exists a d-variate polynomial f of degree O(r 1/d ) such that each connected component of R d \ Z(f ) contains at most n/r points of P , where Z(f ) is the zero set of f . We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications. *

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“…Theorem 2.5 (Barone and Basu [BB12]). Let V be a k-dimensional algebraic variety in R d defined by a finite set F of d-variate real polynomials, each of degree at most D, and let G be a set of s polynomials of degree at most E ≥ D. Then the number of those cells of the arrangement of the zero sets of F ∪ G that are contained in V is bounded by…”

Section: Algebraic Preliminariesmentioning

confidence: 99%

“…Moreover, using a more recent result of Barone and Basu [BB12] discussed below, one obtains that an algebraic variety X of dimension k defined by polynomials of constant-bounded degrees crosses at most O(r k/d ) components of R d \ Z(f ). In this respect, polynomial partitions match the performance of simplicial partitions concerning hyperplanes and give a crucial advantage for other varieties.…”

Section: Introductionmentioning

confidence: 97%

Multilevel Polynomial Partitions and Simplified Range Searching

Matoušek

Patáková

2015

Discrete Comput Geom

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The polynomial partitioning method of Guth and Katz [arXiv:1011.4105] has numerous applications in discrete and computational geometry. It partitions a given n-point set P ⊂ R d using the zero set Z(f ) of a suitable d-variate polynomial f . Applications of this result are often complicated by the problem, what should be done with the points of P lying within Z(f )? A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far it has been pursued with limited success-several authors managed to construct and apply a second partitioning polynomial, but further progress has been prevented by technical obstacles.We provide a polynomial partitioning method with up to d polynomials in dimension d, which allows for a complete decomposition of the given point set. We apply it to obtain a new algorithm for the semialgebraic range searching problem. Our algorithm has running time bounds similar to a recent algorithm by Agarwal, Sharir, and the first author [SIAM J. Comput. 42: 2039-2062, 2013], but it is simpler both conceptually and technically. While this paper has been in preparation, Basu and Sombra, as well as Fox, Pach, Sheffer, Suk, and Zahl, obtained results concerning polynomial partitions which overlap with ours to some extent.

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Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Cited by 44 publications

References 26 publications

An Incidence Theorem in Higher Dimensions

An Incidence Theorem in Higher Dimensions

On Range Searching with Semialgebraic Sets II

Multilevel Polynomial Partitions and Simplified Range Searching

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