E cient Randomized Algorithms for Some GeometricOptimization Problems

Abstract

In this paper we rst prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let F be a collection of n totally or partially deened algebraic trivariate functions of constant maximum degree, with the additional property that, for a given pair of functions f; f 0 2 F, the surface f (x; y; z) = f 0 (x; y; z) is xy-monotone (actually, we need a somewhat weaker property|see below). We show that the vertical decomposition of the minimization diagram of F consists of O(n 3+") cells (each of constant complexity), for any " > 0. In the second part of the paper we present a general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including (i) computing the width of a point set in 3-space, (ii) computing the minimum-width annulus enclosing a set of n points in the plane, and (iii) computing thèbiggest stick' inside a simple polygon in the plane. Using the above result on vertical decompositions, we show that the expected running time of all three algorithms is O(n 3=2+"), for any " > 0. Our algorithm improves and simpliies previous solutions of all three problems.

Cite this paper

@inproceedings{Agarwal1995ECR, title={E cient Randomized Algorithms for Some GeometricOptimization Problems}, author={Pankaj K. Agarwal and Micha Sharir and Pankaj K. Agarwaly and Micha Sharirz}, year={1995} }