Robust Kinetic Convex Hulls in 3D

Kinetic data structures provide a framework for computing combinatorial properties of continuously moving objects. Although kinetic data structures for many problems have been proposed, some difficulties remain in devising and implementing them, especially robustly. One set of difficulties stems from the required update mechanisms used for processing certificate failures-devising efficient update mechanisms can be difficult, especially for sophisticated problems such as those in 3D. Another set of difficulties arises due to the strong assumption in the framework that the update mechanism is invoked with a single event. This assumption requires ordering the events precisely, which is generally expensive. This assumption also makes it difficult to deal with simultaneous events that arise due to degeneracies or due to intrinsic properties of the kinetized algorithms. In this paper, we apply advances on self-adjusting computation to provide a robust motion simulation technique that combines kinetic event-based scheduling and the classic idea of fixed-time sampling. The idea is to divide time into a lattice of fixed-size intervals, and process events at the resolution of an interval. We apply the approach to the problem of kinetic maintenance of convex hulls in 3D, a problem that has been open since 90s. We evaluate the effectiveness of the proposal experimentally. Using the approach, we are able to run simulations consisting of tens of thousands of points robustly and efficiently.