Generating Necklaces

A k color n bead necklace is an equivalence class of k ary n tuples under rotation In this paper we analyze an algorithm due to Fredricksen Kessler and Maiorana FKM to show that necklaces can be generated in constant amortized time We also present a new approach to generating necklaces which we conjecture can also be implemented in constant amortized time The FKM algorithm generates a list of n tuples which includes among other things the lexicographically smallest element of each k color n bead necklace Previously it had been shown only that the list contains at most O n N k n elements where N k n is the number of k color n bead necklaces and that successive elements can be generated in worst case time O n giving a bound of O n N k n on the time for the algorithm We show that the number of elements generated by the FKM algorithm approaches k k N k n and the total time is only O N k n A by product of our analysis is a precise characterization of the list generated by FKM which makes a recursive description possible