Efficient Ranking of Lyndon Words and Decoding Lexicographically Minimal de Bruijn Sequence
Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter · SIAM Journal on Discrete Mathematics · 2016
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 $\mathcal{O}(n)$ 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 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 (a theorem of Fredricksen and Maiorana) to develop the first polynomial-time algorithm for decoding the minimal de Bruijn sequence of any rank $n$ (it determines the position of a given word of length $n$ within the de Bruijn sequence).