Improved lower bounds for the cycle detection problem

Lower bounds for the 'cycle detection problem' were recently investigated by Fich (1981, 1983). 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 (1981, 1983) 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. © 1985.