Algorithms for Computing Abelian Periods of Words

Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the notion of an \emph{Abelian period} of a word. A word of length $n$ over an alphabet of size $\sigma$ can have $\Theta(n^{2})$ distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time $O(n^2 \times \sigma)$ using $O(n \times \sigma)$ space. We present an off-line algorithm based on a \sel function having the same worst-case theoretical complexity as the Brute-Force one, but outperforming it in practice. We then present on-line algorithms that also enable to compute all the Abelian periods of all the prefixes of $w$.Comment: Accepted for publication in Discrete Applied Mathematic

Fast Computation of Abelian Runs

Given a word $w$ and a Parikh vector $\mathcal{P}$, an abelian run of period $\mathcal{P}$ in $w$ is a maximal occurrence of a substring of $w$ having abelian period $\mathcal{P}$. Our main result is an online algorithm that, given a word $w$ of length $n$ over an alphabet of cardinality $\sigma$ and a Parikh vector $\mathcal{P}$, returns all the abelian runs of period $\mathcal{P}$ in $w$ in time $O(n)$ and space $O(\sigma+p)$, where $p$ is the norm of $\mathcal{P}$, i.e., the sum of its components. We also present an online algorithm that computes all the abelian runs with periods of norm $p$ in $w$ in time $O(np)$, for any given norm $p$. Finally, we give an $O(n^2)$-time offline randomized algorithm for computing all the abelian runs of $w$. Its deterministic counterpart runs in $O(n^2\log\sigma)$ time.Comment: To appear in Theoretical Computer Scienc

Counting distinct palindromes in a word in linear time

International audienceWe design an algorithm to count the number of distinct palindromes in a word w in time O(|w|), by adapting an algorithm to detect all occurrences of maximal palindromes in a given word and using the longest previous factor array. As a direct consequence, this shows that the palindromic richness (or fullness) of a word can be checked in linear time

Multiplions et divisons avec des bâtons : Les réglettes de Genaille et Lucas.

http://www.apmep.asso.fr/spip.php?page=article&id_article=3624National audienceLes réglettes de Genaille et Lucas ont été inventées par l'ingénieur Henri Genaille en 1885 sur une proposition du mathématicien Édouard Lucas. Il existe des réglettes pour les multiplications et pour les divisions entières. Ces réglettes furent beaucoup utilisées jusque dans les années 1920, période d'apparition des règles à calcul qui précédèrent les calculatrices. Les réglettes présentées dans ce document permettent d'effectuer rapidement de grandes multiplications ou divisions sans calculer de tête. Nous expliquons comment les manipuler et les construire

Combining MEDLINE and publisher data to create parallel corpora for the automatic translation of biomedical text

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Online Computation of Abelian Runs

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Computing abelian periods in words

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Online Computation of Abelian Runs

To appear in the Proceedings of LATA 2015Given a word $w$ and a Parikh vector $\mathcal{P}$, an abelian run of period $\mathcal{P}$ in $w$ is a maximal occurrence of a substring of $w$ having abelian period $\mathcal{P}$. We give an algorithm that finds all the abelian runs of period $\mathcal{P}$ in a word of length $n$ in time $O(n\times |\mathcal{P}|)$ and space $O(\sigma+|\mathcal{P}|)$

A note on easy and efficient computation of full abelian periods of a word

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