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+epsilon)) for any epsilon > 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+epsilon)), for any epsilon > 0, a data structure of size O(n(3+epsilon)) 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+epsilon)), for any epsilon > 0, improving previous solutions that run in time O(n(8/5+epsilon)). 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+epsilon)) storage and preprocessing time, for any epsilon > 0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.