k-Abelian Pattern Matching

Two words are called $k$-abelian equivalent, if they share the same multiplicities for all factors of length at most $k$. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are $k$-abelian equivalent to some pattern. Moreover, an optimal algorithm for finding the largest $k$ for which two words are $k$-abelian equivalent is given. Solutions for various online versions of the $k$-abelian pattern matching problem are also proposed

Detecting k-(Sub-)Cadences and Equidistant Subsequence Occurrences

The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive symbols of $P$ in the occurrence are equally spaced. We can consider the problem of equidistant subsequences as generalizations of (sub-)cadences. We give bit-parallel algorithms that yield $o(n^2)$ time algorithms for finding $k$-(sub-)cadences and equidistant subsequences. Furthermore, $O(n\log^2 n)$ and $O(n\log n)$ time algorithms, respectively for equidistant and Abelian equidistant matching for the case $|P| = 3$, are shown. The algorithms make use of a technique that was recently introduced which can efficiently compute convolutions with linear constraints

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

On the Parikh-de-Bruijn grid

We introduce the Parikh-de-Bruijn grid, a graph whose vertices are fixed-order Parikh vectors, and whose edges are given by a simple shift operation. This graph gives structural insight into the nature of sets of Parikh vectors as well as that of the Parikh set of a given string. We show its utility by proving some results on Parikh-de-Bruijn strings, the abelian analog of de-Bruijn sequences.Comment: 18 pages, 3 figures, 1 tabl

Identifying all abelian periods of a string in quadratic time and relevant problems

Abelian periodicity of strings has been studied extensively over the last years. In 2006 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.Comment: Accepted in the "International Journal of foundations of Computer Science

Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.Comment: 38 pages, 5 figure; Added referenc