An Unoriented Variation on de Bruijn Sequences

TL;DR

This paper introduces unoriented de Bruijn sequences, which are read both forwards and backwards, and characterizes their existence and construction using Eulerian paths in specialized graphs.

Contribution

It defines the unoriented de Bruijn sequence, establishes conditions for their existence, and provides a construction method via Eulerian paths in unoriented de Bruijn graphs.

Findings

Optimal unoriented de Bruijn sequences exist only for specific k and n values.

Construction of these sequences relies on Eulerian paths in Eulerized unoriented de Bruijn graphs.

Sequences can be generated for all k, n using the described graph-theoretic approach.

Abstract

For positive integers $k,n$, a de Bruijn sequence $B(k,n)$ is a finite sequence of elements drawn from $k$ characters whose subwords of length $n$ are exactly the $k^n$ words of length $n$ on $k$ characters. This paper introduces the unoriented de Bruijn sequence $uB(k,n)$, an analog to de Bruijn sequences, but for which the sequence is read both forwards and backwards to determine the set of subwords of length $n$. We show that nontrivial unoriented de Bruijn sequences of optimal length exist if and only if $k$ is two or odd and $n$ is less than or equal to 3. Unoriented de Bruijn sequences for any $k$, $n$ may be constructed from certain Eulerian paths in Eulerizations of unoriented de Bruijn graphs.