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Tighter bounds and optimal algorithms for all maximal $$\alpha$$-gapped repeats and palindromes. Finding all maximal $$\alpha$$-gapped repeats and palindromes in optimal worst case time on integer alphabets. (English) Zbl 1386.68120
Summary: An $$\alpha$$-gapped repeat $$(\alpha\geq 1)$$ in a word $$w$$ is a factor $$uvu$$ of $$w$$ such that $$| uv|\leq\alpha| u|$$; the two occurrences of $$u$$ are called arms of this $$\alpha$$-gapped repeat. An $$\alpha$$-gapped repeat is called maximal if its arms cannot be extended simultaneously with the same character to the right nor to the left. We show that the number of all maximal $$\alpha$$-gapped repeats occurring in words of length $$n$$ is upper bounded by $$18\alpha n$$. In the case of $$\alpha$$-gapped palindromes, i.e., factors $$uvu^{\top}$$ with $$| uv|\leq\alpha| u|$$, we show that the number of all maximal $$\alpha$$-gapped palindromes occurring in words of length $$n$$ is upper bounded by $$28\alpha n+7n$$. Both upper bounds allow us to construct algorithms finding all maximal $$\alpha$$-gapped repeats and/or all maximal $$\alpha$$-gapped palindromes of a word of length $$n$$ on an integer alphabet of size $$n^{\mathcal{O}(1)}$$ in $$\mathcal O(\alpha n)$$ time. The presented running times are optimal since there are words that have $$\Theta(\alpha n)$$ maximal $$\alpha$$-gapped repeats/palindromes.

##### MSC:
 68R15 Combinatorics on words 68W32 Algorithms on strings
##### Keywords:
combinatorics on words; counting algorithms
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##### References:
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