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(n d+" ), 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(n 3+" ), for any " ? 0, a data structure of size O(n 3+" ) that, given any query point q, can determine in O(log 2 n) time the function(s) of F that attain the envelope at q. As a consequence, we obtain improved algorithmic solutions to many 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(..