Cut-Down de Bruijn Sequences

TL;DR

This paper introduces simplified and efficient algorithms for constructing cut-down de Bruijn sequences for binary and larger alphabets, improving runtime and space complexity while enabling successor-rule generation.

Contribution

It simplifies and accelerates existing binary algorithms, introduces successor-rule methods, and extends construction techniques to larger alphabets with optimal complexity.

Findings

Binary sequence construction runs in O(n) time per symbol.

First successor-rule approach for binary cut-down de Bruijn sequences.

Extended algorithms to k-ary sequences with similar efficiency.

Abstract

A cut-down de Bruijn sequence is a cyclic string of length $L$, where $1 \leq L \leq k^n$, such that every substring of length $n$ appears at most once. Etzion [Theor. Comp. Sci 44 (1986)] gives an algorithm to construct binary cut-down de Bruijn sequences that requires $o(n)$ simple $n$-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to $\mathcal{O}(n)$ time per symbol generated using $\mathcal{O}(n)$ space. We then provide the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for $k>2$ that runs in $\mathcal{O}(n)$ time per symbol using $\mathcal{O}(n)$ space after some initialization. While our $k$-ary algorithm is based on our simplified version of Etzion's binary algorithm, a number of non-trivial adaptations are required to generalize to larger alphabets.