Distributed algorithm for a color assignment on asynchronous rings

We study a version of the beta-assignment problem (Chang and Lee, 1988) on asynchronous 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 an asynchronous ring. Each agent holds a subset of the items, where initially different agents might hold items associated to the same color. We analyze the problem of distributively assigning colors to agents in such a way that (a) each color is assigned to one agent and (b) the number of different colors assigned to each agent is minimum. Since any color assignment requires that 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 on the ring requires a communication complexity of Omega(n middot m) and then we exhibit a polynomial time distributed algorithm with message complexity matching the bound, that determines a color assignment with cost at most (2 + epsi) times the optimal cost, for any 0 < epsi < 1