### Abstract

The paper consists of two major parts. In the first part, we re-examine relative ε-approximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. Wegive a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative ε-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure-spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace range-counting problem in Rd with relative error ε, and show that relative ε-approximations, combined with the shallow partitioning data structures of Matouŝek, yields ef- ficient solutions to this problem. For example, one of our data structures requires linear storage and O(n1+δ) preprocessingtime, for any > 0, and answers a query in time O(ε-n1-1/bd/2c2b log n), for any > 2/bd/2c; the choice of and affects b and the implied constants. Several variants and extensions are also discussed.

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

Title of host publication | Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07 |

Pages | 327-336 |

Number of pages | 10 |

DOIs | |

State | Published - 2007 |

Event | 23rd Annual Symposium on Computational Geometry, SCG'07 - Gyeongju, Korea, Republic of Duration: Jun 6 2007 → Jun 8 2007 |

### Other

Other | 23rd Annual Symposium on Computational Geometry, SCG'07 |
---|---|

Country | Korea, Republic of |

City | Gyeongju |

Period | 6/6/07 → 6/8/07 |

### Fingerprint

### Keywords

- Approximate range queries
- Discrepancy
- Epsilon-approximations
- Halfspaces
- Partition trees
- Queries
- Range
- Range spaces
- VC-dimension

### ASJC Scopus subject areas

- Software
- Geometry and Topology
- Safety, Risk, Reliability and Quality
- Chemical Health and Safety

### Cite this

*Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07*(pp. 327-336) https://doi.org/10.1145/1247069.1247128

**On approximate halfspace range counting and relative epsilon-approximations.** / Aronov, Boris; Har-Peled, Sariel; Sharir, Micha.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07.*pp. 327-336, 23rd Annual Symposium on Computational Geometry, SCG'07, Gyeongju, Korea, Republic of, 6/6/07. https://doi.org/10.1145/1247069.1247128

}

TY - GEN

T1 - On approximate halfspace range counting and relative epsilon-approximations

AU - Aronov, Boris

AU - Har-Peled, Sariel

AU - Sharir, Micha

PY - 2007

Y1 - 2007

N2 - The paper consists of two major parts. In the first part, we re-examine relative ε-approximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. Wegive a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative ε-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure-spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace range-counting problem in Rd with relative error ε, and show that relative ε-approximations, combined with the shallow partitioning data structures of Matouŝek, yields ef- ficient solutions to this problem. For example, one of our data structures requires linear storage and O(n1+δ) preprocessingtime, for any > 0, and answers a query in time O(ε-n1-1/bd/2c2b log n), for any > 2/bd/2c; the choice of and affects b and the implied constants. Several variants and extensions are also discussed.

AB - The paper consists of two major parts. In the first part, we re-examine relative ε-approximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. Wegive a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative ε-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure-spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace range-counting problem in Rd with relative error ε, and show that relative ε-approximations, combined with the shallow partitioning data structures of Matouŝek, yields ef- ficient solutions to this problem. For example, one of our data structures requires linear storage and O(n1+δ) preprocessingtime, for any > 0, and answers a query in time O(ε-n1-1/bd/2c2b log n), for any > 2/bd/2c; the choice of and affects b and the implied constants. Several variants and extensions are also discussed.

KW - Approximate range queries

KW - Discrepancy

KW - Epsilon-approximations

KW - Halfspaces

KW - Partition trees

KW - Queries

KW - Range

KW - Range spaces

KW - VC-dimension

UR - http://www.scopus.com/inward/record.url?scp=35348843977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348843977&partnerID=8YFLogxK

U2 - 10.1145/1247069.1247128

DO - 10.1145/1247069.1247128

M3 - Conference contribution

SN - 1595937056

SN - 9781595937056

SP - 327

EP - 336

BT - Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07

ER -