### Abstract

Let F ℱe 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, for any query point q, can determine in O(log^{2} n) time the function(s) of ℱ that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several 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 randomized expected 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) |
---|---|

Pages (from-to) | 1714-1732 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 26 |

Issue number | 6 |

State | Published - Dec 1997 |

### Fingerprint

### Keywords

- Closest pair
- Lower envelopes
- Point location
- Ray shooting

### ASJC Scopus subject areas

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

### Cite this

*SIAM Journal on Computing*,

*26*(6), 1714-1732.

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

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 26, no. 6, pp. 1714-1732.

}

TY - JOUR

T1 - Computing envelopes in four dimensions with applications

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - Sharir, Micha

PY - 1997/12

Y1 - 1997/12

N2 - Let F ℱe 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, for any query point q, can determine in O(log2 n) time the function(s) of ℱ that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several 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 randomized expected 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 ℱe 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, for any query point q, can determine in O(log2 n) time the function(s) of ℱ that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several 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 randomized expected 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.

KW - Closest pair

KW - Lower envelopes

KW - Point location

KW - Ray shooting

UR - http://www.scopus.com/inward/record.url?scp=0000612565&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000612565&partnerID=8YFLogxK

M3 - Article

VL - 26

SP - 1714

EP - 1732

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 6

ER -