Efficient constructions of the Prefer-same and Prefer-opposite de Bruijn sequences

Abstract

The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert et al. [ AIEE Transactions 77 (1958)]. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length \(2^{n}\) de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this paper, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using \(O(n)\) time per bit and only \(O(n)\) space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.