# De Bruijn Sequences Revisited

### Venue

International Journal of Foundations of Computer Science, vol. 23 (2012), pp. 1307-1322

### Publication Year

2012

### Authors

Lila Kari, Zhi Xu

### BibTeX

## Abstract

A (non-circular) de Bruijn sequence w of order n is a word such that every word of
length n appears exactly once in w as a factor. In this paper, we generalize the
concept to different settings: the multi-shift de Bruijn sequence and the pseudo de
Bruijn sequence. An m-shift de Bruijn sequence of order n is a word such that every
word of length n appears exactly once in w as a factor that starts at a position im
+ 1 for some integer i ≥ 0. A pseudo de Bruijn sequence of order n with respect to
an antimorphic involution θ is a word such that for every word u of length n the
total number of appearances of u and θ(u) as a factor is one. We show that the
number of m-shift de Bruijn sequences of order n is an!a(m-n)(an-1) for 1 ≤ n ≤ m
and is (am!)an-m for 1 ≤ m ≤ n, where a is the size of the alphabet. We provide two
algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de
Bruijn sequence is important for solving the Frobenius problem in a free monoid. We
show that the existence of pseudo de Bruijn sequences depends on the given alphabet
and antimorphic involution, and obtain formulas for the number of such sequences in
some particular settings.