Length Synchronizing word

the problem of estimating length of synchronizing words has long history , posed independently several authors, commonly known Černý conjecture. in 1964 jan Černý conjectured (n − 1) upper bound length of shortest synchronizing word n-state complete dfa (a dfa complete state transition graph). if true, tight: in 1964 paper, Černý exhibited class of automata (indexed number n of states) shortest reset words have length. best upper bound known (n - n)/6, far lower bound.

for n-state dfas on k-letter input alphabet, algorithm david eppstein finds synchronizing word of length @ 11n/48 + o(n), , runs in time complexity o(n+kn). algorithm not find shortest possible synchronizing word given automaton; eppstein shows, problem of finding shortest synchronizing word np-complete. however, special class of automata in state transitions preserve cyclic order of states, describes different algorithm time o(kn) finds shortest synchronizing word, proves these automata have synchronizing word of length @ (n − 1) (the bound given in Černý s conjecture), , exhibits examples of automata special form shortest synchronizing word has length (n − 1).