Lost in Self-Stabilization

Let t(a) and tb a pair of relatively prime positive integers. We work on chains of n(t(a) + t(b)) agents, each of them forming an upper and rightward directed path of the grid Z(2), fromO = (0, 0) toM = (nt(a), nt(b)). We are interested on evolution rules such that, at each time step, an agent is randomly chosen on the chain and is allowed to jump to another site of the grid, with preservation of the connectivity of the chain, and the endpoints. The rules must be local, i. e. the decision of jumping or not only depends on the neighborhood of fixed size s of the randomly chosen agent, and not on the parameters t(a), t(b), n. In the paper, we design such a rule which, starting from any chain which does not crosses the continuous line segment [O, M], reorganizes the chain by iterate applications of the rule, in such a way such that it stabilizes into one of the best possible approximations of [O, M]. The stabilization is reached after O(n(t(a) + t(b)))(4)) iterations.