### Abstract

Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤k-level of the arrangement A(F) is O(k^{3+∈}ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.

Original language | English (US) |
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Title of host publication | Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995 |

Publisher | Association for Computing Machinery |

Pages | 39-50 |

Number of pages | 12 |

Volume | Part F129372 |

ISBN (Electronic) | 0897917243 |

DOIs | |

State | Published - Sep 1 1995 |

Externally published | Yes |

Event | 11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada Duration: Jun 5 1995 → Jun 7 1995 |

### Other

Other | 11th Annual Symposium on Computational Geometry, SCG 1995 |
---|---|

Country | Canada |

City | Vancouver |

Period | 6/5/95 → 6/7/95 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995*(Vol. Part F129372, pp. 39-50). Association for Computing Machinery. https://doi.org/10.1145/220279.220284

**Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications.** / Agarwal, Pankaj K.; Efrat, Alon; Sharir, Micha.

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

*Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995.*vol. Part F129372, Association for Computing Machinery, pp. 39-50, 11th Annual Symposium on Computational Geometry, SCG 1995, Vancouver, Canada, 6/5/95. https://doi.org/10.1145/220279.220284

}

TY - GEN

T1 - Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications

AU - Agarwal, Pankaj K.

AU - Efrat, Alon

AU - Sharir, Micha

PY - 1995/9/1

Y1 - 1995/9/1

N2 - Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤k-level of the arrangement A(F) is O(k3+∈ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.

AB - Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤k-level of the arrangement A(F) is O(k3+∈ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.

UR - http://www.scopus.com/inward/record.url?scp=84958016763&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958016763&partnerID=8YFLogxK

U2 - 10.1145/220279.220284

DO - 10.1145/220279.220284

M3 - Conference contribution

AN - SCOPUS:84958016763

VL - Part F129372

SP - 39

EP - 50

BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995

PB - Association for Computing Machinery

ER -