Universal Shape Formation for Programmable Matter

We envision programmable matter consisting of systems of computationally limited devices (which we call particles) that are able to self-organize in order to achieve a desired collective goal without the need for central control or external intervention. Central problems for these particle systems are shape formation and coating problems. In this paper, we present a universal shape formation algorithm which takes an arbitrary shape composed of a constant number of equilateral triangles of unit size and lets the particles build that shape at a scale depending on the number of particles in the system. Our algorithm runs in O(√n) asynchronous execution rounds, where $n$ is the number of particles in the system, provided we start from a well-initialized configuration of the particles. This is optimal in a sense that for any shape deviating from the initial configuration, any movement strategy would require Ω(√n) rounds in the worst case (over all asynchronous activations of the particles). Our algorithm relies only on local information (e.g., particles do not have ids, nor do they know n, or have any sort of global coordinate system), and requires only a constant-size memory per particle.