Lost in self-stabilization: A local process that aligns connected cells

Let t a and t b be a pair of relatively prime positive integers. We work on chains of n ( t a + t b ) agents which form together an upper and rightward directed path of the grid Z 2 from O = ( 0 , 0 ) to M = ( n t a , n t 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 preserving 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 . Moreover, the parameter s only depends on t a + t b and not on n. In this paper, we design a rule such that, starting from any chain which does not cross the continuous line segment [ O , M ] , this rule reorganizes this chain into one of the best possible approximations of [ O , M ] . The stabilization is reached after O ( ( n ( t a + t b ) ) 4 ) iterations, in average. The work presented here is at the crossroad of many different domains such as modeling a stabilizing process in crystallography, stochastic cellular automata, organizing a line of robots in distributed algorithms (the robot chain problem) and Christoffel words in language theory.