Extending de Bruijn sequences to larger alphabets

https://doi.org/10.1016/j.ipl.2020.106085Get rights and content

Highlights

  • Novel form of extension of de Bruijn sequences.

  • Extension of a de Bruijn sequence to a larger alphabet but of the same order.

  • No long runs without the new symbol.

  • Extension of an Eulerian cycle on an expanded de Bruijn graph.

  • Solution as a problem of maximum flow.

Abstract

A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence where each length-n sequence occurs exactly once. We present a way of extending de Bruijn sequences by adding a new symbol to the alphabet: the extension is performed by embedding a given de Bruijn sequence into another one of the same order, but over the alphabet with one more symbol, while ensuring that there are no long runs without the new symbol. Our solution is based on auxiliary graphs derived from the de Bruijn graph and solving a problem of maximum flow.

Section snippets

Introduction and statement of results

A circular sequence is the equivalence class of a sequence under rotations. A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence of length kn in which every length-n sequence occurs exactly once [6], [11], see [4] for a fine presentation and history. For example, writing [abc] to denote the circular sequence formed by the rotations of abc, [0011] is de Bruijn of order 2 over the alphabet {0,1}.

A subsequence of a sequence a1a2an is a sequence b1b2bm defined by bi=ani

Proof of Theorem 1

In the sequel we use the terms word and sequence interchangeably. A de Bruijn graph G(k,n) is a directed graph whose vertices are the words of length n over a k-symbol alphabet and whose edges are the pairs (v,w) where v=au and w=ub, for some word u of length n1 and possibly two different symbols a,b. Thus, the graph G(k,n) has kn vertices and kn+1 edges, it is strongly connected and every vertex has the same in-degree and out-degree. Each de Bruijn sequence of order n over a k-symbol alphabet

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We are most grateful to one of the referees for their significant recommendations. This research was done under grant Universidad de Buenos Aires UBACyT 20020170100309BA.

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