The partition technique for overlays of envelopes

We obtain a near-tight bound of O(n^3+ε), for any ε > 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 (2000), and substantially improves and simplifies a result previously published by the authors (2002). Our bound has numerous algorithmic and combinatorial applications, some of which are presented in this paper. Our result 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 ℱ of n surfaces is partitioned into k subcollections ℱ_i of n/k surfaces each, and the complexity of the relevant combinatorial structure in ℱ is recursively related to the complexities of the corresponding structures in each of the 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 R³, thereby simplifying the previously available proof (1996).