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

A Note on Efficient Computation of All Abelian Periods in a String

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all `long' Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems

A Note on Easy and Efficient Computation of Full Abelian Periods of a Word

Constantinescu and Ilie (Bulletin of the EATCS 89, 167-170, 2006) introduced the idea of an Abelian period with head and tail of a finite word. An Abelian period is called full if both the head and the tail are empty. We present a simple and easy-to-implement $O(n\log\log n)$-time algorithm for computing all the full Abelian periods of a word of length $n$ over a constant-size alphabet. Experiments show that our algorithm significantly outperforms the $O(n)$ algorithm proposed by Kociumaka et al. (Proc. of STACS, 245-256, 2013) for the same problem.Comment: Accepted for publication in Discrete Applied Mathematic

On minimizing deterministic tree automata

We present two algorithms for minimizing deterministic frontier-to-root tree automata (dfrtas) and compare them with their string counterparts. The presentation is incremental, starting out from definitions of minimality of automata and state equivalence, in the style of earlier algorithm taxonomies by the authors. The first algorithm is the classical one, initially presented by Brainerd in the 1960s and presented (sometimes imprecisely) in standard texts on tree language theory ever since. The second algorithm is completely new. This algorithm, essentially representing the generalization to ranked trees of the string algorithm presented by Watson and Daciuk, incrementally minimizes a dfrta. As a result, intermediate results of the algorithm can be used to reduce the initial automaton’s size. This makes the algorithm useful in situations where running time is restricted (for example, in real-time applications). We also briefly sketch how a concurrent specification of the algorithm in CSP can be obtained from an existing specification for the dfa case

On baier's sort of maximal Lyndon substrings

We describe and analyze in terms of Lyndon words an elementary sort of maximal Lyndon factors of a string and prove formally its correctness. Since the sort is based on the first phase of Baier’s algorithm for sorting of the suffixes of a string, we refer to it as Baier’s sort

On minimizing deterministic tree automata

We present two algorithms for minimizing deterministic frontier-to-root tree automata (dfrtas) and compare them with their string counterparts. The presentation is incremental, starting out from definitions of minimality of automata and state equivalence, in the style of earlier algorithm taxonomies by the authors. The first algorithm is the classical one, initially presented by Brainerd in the 1960s and presented (sometimes imprecisely) in standard texts on tree language theory ever since. The second algorithm is completely new. This algorithm, essentially representing the generalization to ranked trees of the string algorithm presented by Watson and Daciuk, incrementally minimizes a dfrta. As a result, intermediate results of the algorithm can be used to reduce the initial automaton’s size. This makes the algorithm useful in situations where running time is restricted (for example, in real-time applications). We also briefly sketch how a concurrent specification of the algorithm in CSP can be obtained from an existing specification for the dfa case