Arbitrary-length analogs to de Bruijn sequences

TL;DR

This paper introduces an algorithm to generate sequences with properties similar to de Bruijn sequences for any length and alphabet size, extending existing methods to arbitrary lengths efficiently.

Contribution

It presents a novel algorithm that constructs analogs of de Bruijn sequences for arbitrary lengths and alphabet sizes in linear time, extending Lempel's recursive approach.

Findings

Algorithm runs in O(L) time and uses O(L log K) space.

Supports any sequence length L and alphabet size K ≥ 2.

Implementation available in Python at provided URL.

Abstract

Let $\widetilde{\alpha}$ be a length-$L$ cyclic sequence of characters from a size-$K$ alphabet $\mathcal{A}$ such that the number of occurrences of any length-$m$ string on $\mathcal{A}$ as a substring of $\widetilde{\alpha}$ is $\lfloor L / K^m \rfloor$ or $\lceil L / K^m \rceil$. When $L = K^N$ for any positive integer $N$, $\widetilde{\alpha}$ is a de Bruijn sequence of order $N$, and when $L \neq K^N$, $\widetilde{\alpha}$ shares many properties with de Bruijn sequences. We describe an algorithm that outputs some $\widetilde{\alpha}$ for any combination of $K \geq 2$ and $L \geq 1$ in $O(L)$ time using $O(L \log K)$ space. This algorithm extends Lempel's recursive construction of a binary de Bruijn sequence. An implementation written in Python is available at https://github.com/nelloreward/pkl.