The Partition Technique for Overlays of Envelopes

Abstract

We obtain a near-tight bound of $O(n^{3+\varepsilon})$ for any $\varepsilon>0$ on the complexity of the overlay of the minimization diagrams of two collections of surfaces in four dimensions. This settles a long-standing problem in the theory of arrangements, most recently cited by Agarwal and Sharir [in Handbook of Computational Geometry, North--Holland, Amsterdam, 2000, pp. 49--119, Open Problem 2], and substantially improves and simplifies a result previously published by the authors [in Proceedings of the rm13th ACM--SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2002, pp. 810--819]. Our bound is obtained by introducing a new approach to the analysis of combinatorial structures arising in geometric arrangements of surfaces. This approach, which we call the "partition technique," is based on k-fold divide and conquer, in which a given collection ${\cal F}$ of n surfaces is partitioned into k subcollections ${\cal F}_i$ of n/k surfaces each, and the complexity of the relevant combinatorial structure in ${\cal F}$ is recursively related to the complexities of the corresponding structures in each of the ${\cal F}_i$'s. We introduce this approach by applying it first to obtain a new simple proof for the known near-quadratic bound on the complexity of an overlay of two minimization diagrams of collections of surfaces in $\mathbb{R}^3$, thereby simplifying the previously available proof [P. K. Agarwal, O. Schwarzkopf, and M. Sharir, Discrete Comput. Geom., 15 (1996), pp. 1--13]. The main new bound on overlays has numerous algorithmic and combinatorial applications, some of which are presented in this paper.