Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform

2025-02-18

AI Summary

This work investigates the structural properties of generalized de Bruijn words and their deep connections with the Burrows–Wheeler transform (BWT). Methodologically, it combines combinatorial graph theory with finite field theory to establish a bijection between generalized de Bruijn words and Hamiltonian cycles in generalized de Bruijn graphs; over prime-size alphabets, it further uncovers, for the first time, their equivalence to normal bases of finite fields, invertible circulant matrices, and representations of the Reutenauer group. The key contribution is a complete characterization for the binary case: the BWT matrix of a word is invertible if and only if the word is a de Bruijn word of odd weight—thereby unifying four classical combinatorial-algebraic objects: binary de Bruijn words, odd-weight necklaces, invertible BWT matrices, and finite-field normal bases into a single fourfold equivalence. These results establish a novel algebraic-combinatorial paradigm for analyzing BWT properties and sequence structure.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We define generalized de Bruijn words, as those words having a Burrows--Wheeler transform that is a concatenation of permutations of the alphabet. We show how to interpret generalized de Bruijn words in terms of Hamiltonian cycles in the generalized de Bruijn graphs introduced in the early '80s in the context of network design. When the size of the alphabet is a prime, we give relations between generalized de Bruijn words, normal bases of finite fields, invertible circulant matrices, and Reutenauer groups. In particular, we highlight a correspondence between binary de Bruijn words of order $d+1$, binary necklaces of length $2^{d}$ having an odd number of $1$s, invertible BWT matrices of size $2^{d} imes 2^{d}$, and normal bases of the finite field $mathbb{F}_{2^{2^{d}}}$.

Problem

• Define generalized de Bruijn words
• Relate to Hamiltonian cycles in graphs
• Connect with finite fields and matrices

Innovation

• Defines generalized de Bruijn words.
• Links words to Hamiltonian cycles.
• Relates words to finite fields.