# On range searching with semialgebraic sets. II

Journal Article (Journal Article)

Let P be a set of n points in ℝd. 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(n11/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 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 ≤ n, there exists a d-variate polynomial f of degree O(r1/d) such that each connected component of ℝd \ 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. © 2013 Society for Industrial and Applied Mathematics.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Matoušek, J; Sharir, M

### Published Date

- December 23, 2013

### Published In

### Volume / Issue

- 42 / 6

### Start / End Page

- 2039 - 2062

### International Standard Serial Number (ISSN)

- 0097-5397

### Digital Object Identifier (DOI)

- 10.1137/120890855

### Citation Source

- Scopus