The main contribution of this paper is an algorithm to solve an extended version of the quantized consensus problem over networks represented by Hamiltonian graphs, i.e., graphs containing a Hamiltonian cycle, which we assume to be known in advance. Given a network of agents, we assume that a certain number of tokens should be assigned to the agents, so that the total number of tokens weighted by their sizes is the same for all the agents. The algorithm is proved to converge almost surely to a finite set containing the optimal solution. A worst case study of the expected convergence time is carried out, thus proving the efficiency of the algorithm with respect to other solutions recently presented in the literature. Moreover, the algorithm has a decentralized stop criterion once the convergence set is reached. (C) 2011 Elsevier Ltd. All rights reserved.