An Algorithm for Group Formation and Maximal Independent Set in an Amorphous Computer

Amorphous computing is the study of programming ultra-scale computing environments of smart sensors and actuators \cite{white-paper}. The individual elements are identical, asynchronous, randomly placed, embedded and communicate locally via wireless broadcast. Aggregating the processors into groups is a useful paradigm for programming an amorphous computer because groups can be used for specialization, increased robustness, and efficient resource allocation. This paper presents a new algorithm, called the {\em clubs algorithm}, for efficiently aggregating processors into groups in an amorphous computer, in time proportional to the local density of processors. The clubs algorithm is well-suited to the unique characteristics of an amorphous computer. In addition, the algorithm derives two properties from the physical embedding of the amorphous computer: an upper bound on the number of groups formed and a constant upper bound on the density of groups. The clubs algorithm can also be extended to find the maximal independent set (MIS) and $\Delta + 1$ vertex coloring in an amorphous computer in $O(\log N)$ rounds, where $N$ is the total number of elements and $\Delta$ is the maximum degree.