A Faster Algorithm for Computing Maximal $$\alpha $$-gapped Repeats in a String

Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes

“…Finally, in August 2015, the fourth author of this paper announced on the Stringmasters webpage that the bound on the number of all maximal α-gapped repeats and palindromes is indeed O(αn); together with [22], this leads to an optimal algorithm for solving the problem of finding all maximal α-gapped repeats in the particular case of constant alphabets. This announcement was followed by Crochemore et al [7] who confirmed the bound |G α (w)| = O(αn); additionally, they presented an algorithm computing all maximal α-gapped repeats for constant alphabets in O(αn) time.…”

“…Problems like how many maximal α-gapped repeats or palindromes can a word of length n contain, how efficiently can we compute the set of all maximal α-gapped repeats or palindromes in a word, how efficiently can we compute the α-gapped repeat or palindrome with the longest arm, were already investigated [3,7,10,22]: [10] showed how to compute the longest α-gapped repeat/palindrome in O(αn) time, [8] showed how to compute a series of data structures that can give the longest 2-gapped repeat/palindrome that starts at each position (and the results generalize easily to arbitrary α), Tanimura et al [22] gave an O(αn + |G α (w)|)-time solution to find all maximal α-gapped repeats for an input word over constant alphabets. Finally, in August 2015, the fourth author of this paper announced on the Stringmasters webpage that the bound on the number of all maximal α-gapped repeats and palindromes is indeed O(αn); together with [22], this leads to an optimal algorithm for solving the problem of finding all maximal α-gapped repeats in the particular case of constant alphabets.…”

“…Our algorithms require a deeper analysis than the one developed in [10] for finding the longest α-gapped repeats. Asides, they use essentially different techniques and data structures than the ones described in [7,18,22].…”

Efficient Representation and Counting of Antipower Factors in Words

“…The first algorithm for computing α-MGRs was proposed by Kolpakov et al [11]. It was improved by Crochemore et al [6], Tanimura et al [13], and finally Gawrychowski et al [9], who showed the following result. A run (a maximal repetition) in a word w is a triple (i, j, p) such that w[i .…”

Counting maximal-exponent factors in words

“…Note that allowing the two occurrences of u relates to counting runs and the condition implies that the exponent of α-gapped repeats is at least 1 + 1/α. After a more restrictive notion of fix-gapped repeat in [14,17], locating and counting α-gapped repeats was studied first in [7], then more deeply in [16] and in [11,19]. Eventually, algorithms to locate α-gapped repeats optimally in time O(αn) are described in [9,13].…”