### Abstract

We present a simple scheme extending the shallow partitioning data structures of Matoušek, which supports efficient approximate halfspace range-counting queries in ℝ
^{d} with relative error ε. Specifically, the problem is, given a set P of n points in ℝ
^{d}, to preprocess them into a data structure that returns, for a query halfspace h, a number t so that (1 - ε)|h ∩ P| ≤ t ≤ (1 + ε)| h ∩P|. One of our data structures requires linear storage and O(n
^{1+δ}) preprocessing time, for any δ > 0, and answers a query in time O(ε
^{-γ} n
^{1-1/⌊d/2⌋}2
^{b} log
^{*} n) for any γ > 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step, and so are the query, storage, and preprocessing costs. The quality of approximation, for every query, is guaranteed with high probability. The construction can also be fully derandomized, at the expense of increasing preprocessing time.

Original language | English (US) |
---|---|

Pages (from-to) | 2704-2725 |

Number of pages | 22 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 7 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Approximation algorithms
- Cuttings
- Geometric algorithms
- Geometric sampling
- Hyperplane arrangements
- Partition trees
- Range counting
- Range searching
- Relative approximations
- Shallow cuttings
- Shallow partition trees

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science(all)

### Cite this

*SIAM Journal on Computing*,

*39*(7), 2704-2725. https://doi.org/10.1137/080736600

**Approximate halfspace range counting.** / Aronov, Boris; Sharir, Micha.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 39, no. 7, pp. 2704-2725. https://doi.org/10.1137/080736600

}

TY - JOUR

T1 - Approximate halfspace range counting

AU - Aronov, Boris

AU - Sharir, Micha

PY - 2010

Y1 - 2010

N2 - We present a simple scheme extending the shallow partitioning data structures of Matoušek, which supports efficient approximate halfspace range-counting queries in ℝ d with relative error ε. Specifically, the problem is, given a set P of n points in ℝ d, to preprocess them into a data structure that returns, for a query halfspace h, a number t so that (1 - ε)|h ∩ P| ≤ t ≤ (1 + ε)| h ∩P|. One of our data structures requires linear storage and O(n 1+δ) preprocessing time, for any δ > 0, and answers a query in time O(ε -γ n 1-1/⌊d/2⌋2 b log * n) for any γ > 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step, and so are the query, storage, and preprocessing costs. The quality of approximation, for every query, is guaranteed with high probability. The construction can also be fully derandomized, at the expense of increasing preprocessing time.

AB - We present a simple scheme extending the shallow partitioning data structures of Matoušek, which supports efficient approximate halfspace range-counting queries in ℝ d with relative error ε. Specifically, the problem is, given a set P of n points in ℝ d, to preprocess them into a data structure that returns, for a query halfspace h, a number t so that (1 - ε)|h ∩ P| ≤ t ≤ (1 + ε)| h ∩P|. One of our data structures requires linear storage and O(n 1+δ) preprocessing time, for any δ > 0, and answers a query in time O(ε -γ n 1-1/⌊d/2⌋2 b log * n) for any γ > 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step, and so are the query, storage, and preprocessing costs. The quality of approximation, for every query, is guaranteed with high probability. The construction can also be fully derandomized, at the expense of increasing preprocessing time.

KW - Approximation algorithms

KW - Cuttings

KW - Geometric algorithms

KW - Geometric sampling

KW - Hyperplane arrangements

KW - Partition trees

KW - Range counting

KW - Range searching

KW - Relative approximations

KW - Shallow cuttings

KW - Shallow partition trees

UR - http://www.scopus.com/inward/record.url?scp=77956022804&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956022804&partnerID=8YFLogxK

U2 - 10.1137/080736600

DO - 10.1137/080736600

M3 - Article

VL - 39

SP - 2704

EP - 2725

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 7

ER -