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

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...