Computing Envelopes in Four Dimensions with Applications

Abstract

Let ${\cal F}$ 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 ${\cal F}$ 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(log2n) time the function(s) of ${\cal F}$ 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.