### Abstract

Let F be a collection of n d-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n^{d+ε}), for any ε > 0. For d = 3, by combining this algorithm with the point location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n^{3+ε}), for any ε > 0, a data structure of size O(n^{3+ε}) that, given any query point q, can determine in O(log^{2} n) time whether q lies above, below or on the envelope. As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in time O(n^{17/11+ε}), for any ε > 0, improving previous solutions that run in time O(n^{8/5+ε}). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane, and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n^{3+ε}) storage and preprocessing time, for any ε > 0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

Original language | English (US) |
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Title of host publication | Proceedings of the Annual Symposium on Computational Geometry |

Editors | Anon |

Publisher | Publ by ACM |

Pages | 348-358 |

Number of pages | 11 |

ISBN (Print) | 0897916484 |

State | Published - 1994 |

Event | Proceedings of the 10th Annual Symposium on Computational Geometry - Stony Brook, NY, USA Duration: Jun 6 1994 → Jun 8 1994 |

### Other

Other | Proceedings of the 10th Annual Symposium on Computational Geometry |
---|---|

City | Stony Brook, NY, USA |

Period | 6/6/94 → 6/8/94 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Proceedings of the Annual Symposium on Computational Geometry*(pp. 348-358). Publ by ACM.

**Computing envelopes in four dimensions with applications.** / Agarwal, Pankaj K.; Aronov, Boris; Sharir, Micha.

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

*Proceedings of the Annual Symposium on Computational Geometry.*Publ by ACM, pp. 348-358, Proceedings of the 10th Annual Symposium on Computational Geometry, Stony Brook, NY, USA, 6/6/94.

}

TY - GEN

T1 - Computing envelopes in four dimensions with applications

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - Sharir, Micha

PY - 1994

Y1 - 1994

N2 - Let F be a collection of n d-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(nd+ε), for any ε > 0. For d = 3, by combining this algorithm with the point location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n3+ε), for any ε > 0, a data structure of size O(n3+ε) that, given any query point q, can determine in O(log2 n) time whether q lies above, below or on the envelope. As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in time O(n17/11+ε), for any ε > 0, improving previous solutions that run in time O(n8/5+ε). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane, and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n3+ε) storage and preprocessing time, for any ε > 0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

AB - Let F be a collection of n d-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(nd+ε), for any ε > 0. For d = 3, by combining this algorithm with the point location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n3+ε), for any ε > 0, a data structure of size O(n3+ε) that, given any query point q, can determine in O(log2 n) time whether q lies above, below or on the envelope. As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in time O(n17/11+ε), for any ε > 0, improving previous solutions that run in time O(n8/5+ε). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane, and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n3+ε) storage and preprocessing time, for any ε > 0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

UR - http://www.scopus.com/inward/record.url?scp=0028124028&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028124028&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0028124028

SN - 0897916484

SP - 348

EP - 358

BT - Proceedings of the Annual Symposium on Computational Geometry

A2 - Anon, null

PB - Publ by ACM

ER -