8,695 research outputs found
k-Abelian Pattern Matching
Two words are called -abelian equivalent, if they share the same multiplicities for all factors of length at most . We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are -abelian equivalent to some pattern. Moreover, an optimal algorithm for finding the largest for which two words are -abelian equivalent is given. Solutions for various online versions of the -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 and a text string , we say that is an
\emph{equidistant subsequence} of if is a subsequence of the text such
that consecutive symbols of 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 time
algorithms for finding -(sub-)cadences and equidistant subsequences.
Furthermore, and time algorithms, respectively for
equidistant and Abelian equidistant matching for the case , 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 over an
alphabet of size can have distinct Abelian periods.
The Brute-Force algorithm computes all the Abelian periods of a word in time
using 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 .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
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