On Range Searching with Semialgebraic Sets. II

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. 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 searchin...

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Bibliographic Details
Published in:SIAM journal on computing 2013-01, Vol.42 (6), p.2039-2062
Main Authors: Agarwal, Pankaj K., Matoušek, Jiří, Sharir, Micha
Format: Article
Language:English
Subjects:
Algorithms
Applied mathematics
Computation
Computer science
Data structures
Geometry
Grants
Mathematical models
Partitioning
Polynomials
Queries
Searching
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Summary:Let $P$ be a set of $n$ points in $\mathbb{R}^d$. 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 $d\ge 5$, 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 a partitioning technique of Guth and Katz [On the Erdos distinct distances problem in the plane, arXiv:1011.4105, 2010], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\mathbb{R}^d\setminus Z(f)$ 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. [PUBLICATION ABSTRACT]
ISSN:0097-5397
1095-7111
DOI:10.1137/120890855