Efficient Ranking of Lyndon Words and Decoding Lexicographically Minimal de Bruijn Sequence

TL;DR

This paper introduces efficient algorithms for ranking Lyndon words and decoding minimal de Bruijn sequences, improving computational complexity and enabling practical applications in combinatorics and sequence analysis.

Contribution

It presents the first polynomial-time algorithm for decoding minimal de Bruijn sequences and improves Lyndon word ranking algorithms on the word-RAM model.

Findings

Lyndon word ranking algorithm runs in O(n^2 log σ) time.

Decoding minimal de Bruijn sequences is achieved in polynomial time.

Algorithms are efficient for large alphabet sizes and word lengths.

Abstract

We give efficient algorithms for ranking Lyndon words of length $n$ over an alphabet of size $\sigma$. The rank of a Lyndon word is its position in the sequence of lexicographically ordered Lyndon words of the same length. The outputs are integers of exponential size, and complexity of arithmetic operations on such large integers cannot be ignored. Our model of computations is the word-RAM, in which basic arithmetic operations on (large) numbers of size at most $\sigma^n$ take $O(n)$ time. Our algorithm for ranking Lyndon words makes $O(n^2)$ arithmetic operations (this would imply directly cubic time on word-RAM). However, using an algebraic approach we are able to reduce the total time complexity on the word-RAM to $O(n^2 \log \sigma)$. We also present an $O(n^3 \log^2 \sigma)$-time algorithm that generates the Lyndon word of a given length and rank in lexicographic order. Finally we use the connections between Lyndon words and lexicographically minimal de Bruijn sequences (theorem of Fredricksen and Maiorana) to develop the first polynomial-time algorithm for decoding minimal de Bruijn sequence of any rank $n$ (it determines the position of an arbitrary word of length $n$ within the de Bruijn sequence).