State-transitive Markov processes, random walks on groups, and self-organizing data structures

We prove the optimality of the move-to-front rule in the class of on-line selforganizing sequential search heuristics for data structures, taking advantage of the fact that the sequence of requested items is a Markov chain F whose symmetry group GF acts transitively on the set S of items (F is then said to be state-transitive ). We state corresponding general results on controlled Markov chains with non trivial symmetry group. In order to have a better knowledge of state-transitive Markov processes, we give a construction of any state-transitive Markov process F with the help of a random walk on its symmetry group GF, and a characterization of state-transitive Markov processes F such that their state space S can be endowed with a group structure implying F is a random walk on the group S.