Algorithms for Galois Words: Detection, Factorization, and Rotation
arxiv(2024)
摘要
Lyndon words are extensively studied in combinatorics on words – they play a
crucial role on upper bounding the number of runs a word can have [Bannai+,
SIAM J. Comput.'17]. We can determine Lyndon words, factorize a word into
Lyndon words in lexicographically decreasing order, and find the Lyndon
rotation of a word, all in linear time within constant additional working
space. A recent research interest emerged from the question of what happens
when we change the lexicographic order, which is at the heart of the definition
of Lyndon words. In particular, the alternating order, where the order of all
odd positions becomes reversed, has been recently proposed. While a Lyndon word
is, among all its cyclic rotations, the smallest one with respect to the
lexicographic order, a Galois word exhibits the same property by exchanging the
lexicographic order with the alternating order. Unfortunately, this exchange
has a large impact on the properties Galois words exhibit, which makes it a
nontrivial task to translate results from Lyndon words to Galois words. Up
until now, it has only been conjectured that linear-time algorithms with
constant additional working space in the spirit of Duval's algorithm are
possible for computing the Galois factorization or the Galois rotation.
Here, we affirm this conjecture as follows. Given a word T of length n,
we can determine whether T is a Galois word, in O(n) time with constant
additional working space. Within the same complexities, we can also determine
the Galois rotation of T, and compute the Galois factorization of T online.
The last result settles Open Problem 1 in [Dolce et al., TCS'2019] for Galois
words.
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