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We consider the problem of organizing a scattered group of $n$ robots intwo-dimensional space, with geometric maximum distance $D$ between robots. Thecommunication graph of the swarm is connected, but there is no centralauthority for organizing it. We want to arrange them into a sorted andequally-spaced array between the robots with lowest and highest label, whilemaintaining a connected communication network.  In this paper, we describe a distributed method to accomplish these goals,without using central control, while also keeping time, travel distance andcommunication cost at a minimum. We proceed in a number of stages (leaderelection, initial path construction, subtree contraction, geometricstraightening, and distributed sorting), none of which requires a centralauthority, but still accomplishes best possible parallelization. The overallarraying is performed in $O(n)$ time, $O(n^2)$ individual messages, and $O(nD)$travel distance. Implementation of the sorting and navigation use communicationmessages of fixed size, and are a practical solution for large populations oflow-cost robots.

A Parallel Distributed Strategy for Arraying a Scattered Robot Swarm