A Distributed Message-Optimal Assignment on Rings

Consider a set of items and a set of $m$ colors, where each item is associated to one color. Consider also $n$ computational agents connected by a ring. Each agent holds a subset of the items and items of the same color can be held by different agents. We analyze the problem of distributively assigning colors to agents in such a way that (a) each color is assigned to one agent only and (b) the number of different colors assigned to each agent is minimum. Since any color assignment requires the items be distributed according to it (e.g. all items of the same color are to be held by only one agent), we define the cost of a color assignment as the amount of items that need to be moved, given an initial allocation. We first show that any distributed algorithm for this problem requires a message complexity of $Ω(n\cdot m)$ and then we exhibit an optimal message complexity algorithm for synchronous rings that in polynomial time determines a color assignment with cost at most three times the optimal. We also discuss solutions for the asynchronous setting. Finally, we show how to get a better cost solution at the expenses of either the message or the time complexity.