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...

• Type: Conference
• Year: 1994
• Where: COMPGEOM
• Authors: Pankaj K. Agarwal, Boris Aronov, Micha Sharir
• Keywords: COMPGEOM 1994, Discrete Geometry, Expected Time, Lower Envelopes, Randomized Expected Time