# Distributed Pattern Formation in a Ring

Motivated by concerns about diversity in social networks, we consider the following pattern formation problems in rings. Assume n mobile agents are located at the nodes of an n-node ring network. Each agent is assigned a colour from the set {c_1, c_2, ..., c_q }. The ring is divided into k contiguous blocks or neighbourhoods of length p. The agents are required to rearrange themselves in a distributed manner to satisfy given diversity requirements: in each block j and for each colour c_i, there must be exactly n_i(j) >0 agents of colour c_i in block j. Agents are assumed to be able to see agents in adjacent blocks, and move to any position in adjacent blocks in one time step. When the number of colours q=2, we give an algorithm that terminates in time N_1/n^*_1 + k + 4 where N_1 is the total number of agents of colour c_1 and n^*_1 is the minimum number of agents of colour c_1 required in any block. When the diversity requirements are the same in every block, our algorithm requires 3k+4 steps, and is asymptotically optimal. Our algorithm generalizes for an arbitrary number of colours, and terminates in O(nk) steps. We also show how to extend it to achieve arbitrary specific final patterns, provided there is at least one agent of every colour in every pattern.

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