A computational framework for incremental motion

We propose a generic computational framework for maintaining a discrete geometric structure defined by a collection of static and mobile objects. We assume that the mobile objects move incrementally, that is, in discrete time steps. We assume that the structure to be maintained is a function of the current locations of the mobile and static objects (independent of their prior motion). Unlike other models for kinetic computation, we place no restrictions on the motion nor on its predictability. In order to handle unrestricted incremental motion, our framework is based on the coordination of two computational entities. The first is the incremental motion algorithm. It is responsible for maintaining the structure and a set of certificates, or conditions, that prove the structure's correctness. The other entity, called the motion processor, is responsible for handling all the low-level aspects of motion, including computing and/or tracking the motion of the mobile objects, answering queries about their current positions and velocities, and validating that the object motions satisfy simple motion estimates, which are generated by the incremental motion algorithm. Computational efficiency is measured in terms of the number of interactions between these two entities. We present a simple online protocol, through which the incremental motion algorithm generates motion estimates. We show that, subject to a few assumptions, given a parameter ε > 0, the number of motion estimates generated by this protocol has a competitive ratio of O((log ρ) + 1/ε) relative to an optimal algorithm that has full knowledge of the future motion of the points, but is required to maintain an additional ε-factor separation relative to the certificate failure regions, and has a ratio ρ between its largest and smallest estimates of the motion's drift error.