### 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) |
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Pages (from-to) | 1714-1732 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 26 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1997 |

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### Keywords

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

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

### Cite this

*SIAM Journal on Computing*,

*26*(6), 1714-1732. https://doi.org/10.1137/S0097539794265724