https://hal.archives-ouvertes.fr/hal-00836957Christou, MichalisMichalisChristouInformatics - King‘s College LondonCrochemore, MaximeMaximeCrochemoreInformatics - King‘s College LondonLIGM - Laboratoire d'Informatique Gaspard-Monge - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech - ESIEE Paris - Fédération de Recherche Bézout - CNRS - Centre National de la Recherche ScientifiqueIliopoulos, CostasCostasIliopoulosInformatics - King‘s College LondonIdentifying all abelian periods of a string in quadratic time and relevant problemsHAL CCSD2012stringsalgorithmsabelian periods[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS]Crochemore, Maxime2013-06-21 18:16:232022-09-29 14:21:152013-06-21 18:16:23enJournal articles10.1142/S01290541125001901Abelian periodicity of strings has been studied extensively over the last years. Lately, Constantinescu and Ilie defined the abelian period of a string and several algorithms for the computation of all abelian periods of a string were given. In contrast to the classical period of a word, its abelian version is more flexible, factors of the word are considered the same under any internal permutation of their letters.We show two O(|y|2) algorithms for the computation of all abelian periods of a string y. The first one maps each letter to a suitable number such that each factor of the string can be identified by the unique sum of the numbers corresponding to its letters and hence abelian periods can be identified easily. The other one maps each letter to a prime number such that each factor of the string can be identified by the unique product of the numbers corresponding to its letters and so abelian periods can be identified easily. We also define weak abelian periods on strings and give an O(|y|log(|y|)) algorithm for their computation, together with some other algorithms for more basic problems.