## Abstract

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 semi algebraic 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 ef?cient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.

Original language | English |
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Article number | 6375320 |

Pages (from-to) | 420-429 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 2012 |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: 20 Oct 2012 → 23 Oct 2012 |

## Keywords

- Range searching
- ham-sandwich cuts
- polynomial partition
- semialgebraic sets