Cut-down de Bruijn sequences

Abstract

A cut-down de Bruijn sequence is a cyclic string of length L, where 1≤L≤kn, such that every substring of length n appears at most once. Etzion [Theor. Comp. Sci 44 (1986)] introduced an algorithm to construct binary cut-down de Bruijn sequences requiring o(n) simple n-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to O(n) time per symbol generated using O(n) space. Additionally, we develop the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking/unranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for k>2 that runs in O(n) time per symbol using O(n) space after some initialization.