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Group control and kernels: the 1-d equigrouping problem

One of the canonical problems of group control is to find local rules through which agents construct specified configurations from arbitrary initial positions. In this paper, we introduce and provide several solutions to the 1-d equigrouping problem, a simple but instructive version of the general spatial configuration problem. We show how deterministic solutions are possible on the linear lattice but not the circle, while the reverse is the case for simple probabilistic solutions. We determine a lower bound on the amount and type of information required by any solution, and relate this information to the geometry of the underlying lattice. Finally, we introduce a concept of an interaction kernel, a tool for investigating algorithms in depth. We use the kernel theory to derive several general facts that characterize the group behavior of all deterministic equigrouping solutions, providing a theoretical framework for algorithm design and analysis that may generalize to more complex group control problems.