Randomized Pattern Formation Algorithm for Asynchronous Oblivious Mobile Robots

Abstract

We present a randomized pattern formation algorithm for asynchronous oblivious (i.e., memory-less) mobile robots that enables formation of any target pattern. As for deterministic pattern formation algorithms, the class of patterns formable from an initial configuration I is characterized by the symmetricity (i.e., the order of rotational symmetry) of I, and in particular, every pattern is formable from I if its symmetricity is 1. The randomized pattern formation algorithm ψ _PF we present in this paper consists of two phases: The first phase transforms a given initial configuration I into a configuration I′ such that its symmetricity is 1, and the second phase invokes a deterministic pattern formation algorithm ψ _CWM by Fujinaga et al. (DISC 2012) for asynchronous oblivious mobile robots to finally form the target pattern.

There are two hurdles to overcome to realize ψ _PF . First, all robots must simultaneously stop and agree on the end of the first phase, to safely start the second phase, since the correctness of ψ _CWM is guaranteed only for an initial configuration in which all robots are stationary. Second, the sets of configurations in the two phases must be disjoint, so that even oblivious robots can recognize which phase they are working on. We provide a set of tricks to overcome these hurdles.

This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Molecular Robotics” (No. 24104003 and No. 24104519) of The Ministry of Education, Culture, Sports, Science, and Technology, Japan.