Improved lower bounds for the cycle detection problem (Q1058851)

Lower bounds for the 'cycle detection problem' were recently investigated by \textit{F. E. Fich} [J. Comput. Syst. Sci. 26, 392-409 (1983; Zbl 0509.68040)]. She showed that Floyd's algorithm was optimal among those algorithms which have \(M=2\) memory locations and which make a finite number of 'jumps'. A lower bound for the case where \(M>2\) was also presented, but the question of whether having more than two memory locations could actually yield a better algorithm was left open. In this report, we show that it cannot. A lower bound was also presented by Fich [loc. cit.] for algorithms which have two memory locations and which make a finite number of 'back advances'. We show here that the same lower bound holds even if the restriction on back advances is dropped.