Subquadratic-Time Algorithms for Abelian Stringology Problems

We propose the first subquadratic-time algorithms to a number of natural problems in abelian pattern matching also called jumbled pattern matching for strings over a constant-sized alphabet. Two strings are considered equivalent in this model if the numbers of occurrences of respective symbols in both of them, specified by their so-called Parikh vectors, are the same. We propose the following algorithms for a string of length n:Counting and finding longest/shortest abelian squares in $$On^2/\log ^2n$$ time. Abelian squares were first considered by Erdos 1961; Cummings and Smyth 1997 proposed an $$On^2$$-time algorithm for computing them.Computing all shortest general abelian periods in $$On^2/\sqrt{\log n}$$ time. Abelian periods were introduced by Constantinescu and Ilie 2006 and the previous, quadratic-time algorithms for their computation were given by Fici et al. 2011 for a constant-sized alphabet and by Crochemore et al. 2012 for a general alphabet.Finding all abelian covers in $$On^2/\log n$$ time. Abelian covers were defined by Matsuda et al. 2014.Computing abelian border array in $$On^2/\log ^2n$$ time. This work can be viewed as a continuation of a recent very active line of research on subquadratic space and time jumbled indexing for binary and constant-sized alphabets e.g., Moosa and Rahman, 2012. All our algorithms work in linear space.