Multi-Shift de Bruijn Sequence

TL;DR

This paper introduces a generalized multi-shift de Bruijn sequence, providing formulas for their counts and algorithms for generation, with applications in solving the Frobenius problem in free monoids.

Contribution

It generalizes de Bruijn sequences to a multi-shift setting, deriving formulas for their number and presenting algorithms for their construction.

Findings

Number of multi-shift de Bruijn sequences derived

Algorithms for generating multi-shift de Bruijn sequences provided

Application in solving Frobenius problem in free monoids

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 a multi-shift setting: a multi-shift de Bruijn sequence tau(m,n) of shift m and order n is a word such that every word of length n appears exactly once in w as a factor that starts at index im+1 for some integer i>=0. We show the number of the multi-shift de Bruijn sequence tau(m,n) is (a^n)!a^{(m-n)(a^n-1)} for 1<=n<=m and is (a^m!)^{a^{n-m}} for 1<=m<=n, where a=|Sigma|. We provide two algorithms for generating a tau(m,n). The multi-shift de Bruijn sequence is important in solving the Frobenius problem in a free monoid.