## Computing envelopes in four dimensions with applications (1997)

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Venue: | SIAM J. Comput |

Citations: | 41 - 19 self |

### BibTeX

@ARTICLE{Agarwal97computingenvelopes,

author = {Pankaj K. Agarwal and Boris Aronov and Micha Sharir},

title = {Computing envelopes in four dimensions with applications},

journal = {SIAM J. Comput},

year = {1997},

volume = {26},

pages = {348--358}

}

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### Abstract

Abstract. Let F be a collection of nd-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.