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

TL;DR

This paper introduces efficient algorithms for constructing Prefer-same and Prefer-opposite de Bruijn sequences, significantly reducing space and time complexity from exponential to linear, solving longstanding open problems.

Contribution

It unveils the structure of Prefer-same sequences and provides the first linear-time, linear-space algorithms for their construction and for Prefer-opposite sequences.

Findings

Constructed Prefer-same de Bruijn sequences in O(n) time and space.

Developed an efficient algorithm for Prefer-opposite sequences.

Solved a 60-year open problem in de Bruijn sequence construction.

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.