2 research outputs found
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 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).Comment: Improved version of a paper presented at CPM 201