The Partition Technique for Overlays of Envelopes

We obtain a near-tight bound of O(n3+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 [3, Open Problem 2], and substantially improves and simplifies a result previously published by the authors [17]. 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 F of n surfaces is partitioned into k subcollections Fi 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 Fi’s. We introduce this approach by applying it first to obtain a new simple proof for the kno...

• Type: Conference
• Year: 2002
• Where: FOCS
• Authors: Vladlen Koltun, Micha Sharir
• Keywords: Combinatorial Structure, FOCS 2002, Minimization Diagrams, Surfaces, Theoretical Computer Science