The partition technique for overlays of envelopes

Abstract

We obtain a near-tight bound of O(n/sup 3+/spl epsiv//), for any /spl epsiv/ > 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 /spl Fscr/ of n surfaces is partitioned into k subcollections /spl Fscr//sub i/ of n/k surfaces each, and the complexity of the relevant combinatorial structure in /spl Fscr/ is recursively related to the complexities of the corresponding structures in each of the /spl Fscr//sub 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 /spl Ropf//sup 3/, thereby simplifying the previously available proof (1996).