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COMPGEOM
1994
ACM

Computing Envelopes in Four Dimensions with Applications

13 years 7 months ago
Computing Envelopes in Four Dimensions with Applications
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+) for any > 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+), for any > 0, a data structure of size O(n3+) 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 tim...
Pankaj K. Agarwal, Boris Aronov, Micha Sharir
Added 09 Aug 2010
Updated 09 Aug 2010
Type Conference
Year 1994
Where COMPGEOM
Authors Pankaj K. Agarwal, Boris Aronov, Micha Sharir
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