Improved bounds for the firing synchronization problem

In this pa per we prese nt improve d bounds on the comple xity of solutions to the ¿ ring synchroniza tion problem. In the ¿ ring synchronization proble m we cons ider a one-dime ns ional a rray of n identica l ¿ nite a utoma ta . Initia lly all a utoma ta a re in the same s ta te e xcept for one automaton which is des igna te d a s the initia tor for the synchroniza tion. Our re sults hold for the origina l problem, whe re the initia tor may be located a t eithe r e ndpoint, and for the va ria nt whe re any one of the automata ma y be the initiator, ca lle d the ge nera lize d problem. In both ca ses , the goa l is to de ¿ ne the set of s ta te s a nd trans ition rule s for the a utoma ta so tha t a ll machine s e nte r a specia l ¿ re s tate s imultane ous ly and for the ¿ rs t time during the ¿ na l round of the computation. In our work we improve the cons truction for the bes t known minima l-time solution to the ge nera lize d problem by re ducing the number of s ta te s nee ded and give non-minima l time solutions to the original and ge nera lize d proble m which use fewer s tates tha n the corresponding minima l-time solutions .