# On range searching with semialgebraic sets II

Published

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 semi algebraic sets as ranges, in time close to O(n1-1/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(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 ef?cient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications. © 2012 IEEE.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Matoušek, J; Sharir, M

### Published Date

- December 1, 2012

### Published In

### Start / End Page

- 420 - 429

### International Standard Serial Number (ISSN)

- 0272-5428

### Digital Object Identifier (DOI)

- 10.1109/FOCS.2012.32

### Citation Source

- Scopus