### 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+ε} ψ(n/k)) for any ε > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This bound is nearly optimal in the worst case and implies the existence of shallow cuttings, in the sense of [J. Matousek, Comput. Geom., 2 (1992), pp. 169-186], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including (i) data structures for several generalized 3-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) |
---|---|

Pages (from-to) | 912-953 |

Number of pages | 42 |

Journal | SIAM Journal on Computing |

Volume | 29 |

Issue number | 3 |

State | Published - Dec 1999 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Journal on Computing*,

*29*(3), 912-953.

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

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 29, no. 3, pp. 912-953.

}

TY - JOUR

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

AU - Agarwal, Pankaj K.

AU - Efrat, Alon

AU - Sharir, Micha

PY - 1999/12

Y1 - 1999/12

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+ε ψ(n/k)) for any ε > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This bound is nearly optimal in the worst case and implies the existence of shallow cuttings, in the sense of [J. Matousek, Comput. Geom., 2 (1992), pp. 169-186], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including (i) data structures for several generalized 3-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+ε ψ(n/k)) for any ε > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This bound is nearly optimal in the worst case and implies the existence of shallow cuttings, in the sense of [J. Matousek, Comput. Geom., 2 (1992), pp. 169-186], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including (i) data structures for several generalized 3-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=0033296344&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033296344&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033296344

VL - 29

SP - 912

EP - 953

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 3

ER -