24,491 research outputs found
Searching of gapped repeats and subrepetitions in a word
A gapped repeat is a factor of the form where and are nonempty
words. The period of the gapped repeat is defined as . The gapped
repeat is maximal if it cannot be extended to the left or to the right by at
least one letter with preserving its period. The gapped repeat is called
-gapped if its period is not greater than . A
-subrepetition is a factor which exponent is less than 2 but is not
less than (the exponent of the factor is the quotient of the length
and the minimal period of the factor). The -subrepetition is maximal if
it cannot be extended to the left or to the right by at least one letter with
preserving its minimal period. We reveal a close relation between maximal
gapped repeats and maximal subrepetitions. Moreover, we show that in a word of
length the number of maximal -gapped repeats is bounded by
and the number of maximal -subrepetitions is bounded by
. Using the obtained upper bounds, we propose algorithms for
finding all maximal -gapped repeats and all maximal
-subrepetitions in a word of length . The algorithm for finding all
maximal -gapped repeats has time complexity for the case
of constant alphabet size and time complexity for the
general case. For finding all maximal -subrepetitions we propose two
algorithms. The first algorithm has time
complexity for the case of constant alphabet size and time complexity for the general case. The
second algorithm has
expected time complexity
Variants of Constrained Longest Common Subsequence
In this work, we consider a variant of the classical Longest Common
Subsequence problem called Doubly-Constrained Longest Common Subsequence
(DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings,
and a function Co from A to N, the DC-LCS problem consists in finding the
longest subsequence s of s1 and s2 such that s is a supersequence of all the
strings in Cs and such that the number of occurrences in s of each symbol a in
A is upper bounded by Co(a). The DC-LCS problem provides a clear mathematical
formulation of a sequence comparison problem in Computational Biology and
generalizes two other constrained variants of the LCS problem: the Constrained
LCS and the Repetition-Free LCS. We present two results for the DC-LCS problem.
First, we illustrate a fixed-parameter algorithm where the parameter is the
length of the solution. Secondly, we prove a parameterized hardness result for
the Constrained LCS problem when the parameter is the number of the constraint
strings and the size of the alphabet A. This hardness result also implies the
parameterized hardness of the DC-LCS problem (with the same parameters) and its
NP-hardness when the size of the alphabet is constant
Repetition Detection in a Dynamic String
A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings.
We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer
Since human randomness production has been studied and widely used to assess
executive functions (especially inhibition), many measures have been suggested
to assess the degree to which a sequence is random-like. However, each of them
focuses on one feature of randomness, leading authors to have to use multiple
measures. Here we describe and advocate for the use of the accepted universal
measure for randomness based on algorithmic complexity, by means of a novel
previously presented technique using the the definition of algorithmic
probability. A re-analysis of the classical Radio Zenith data in the light of
the proposed measure and methodology is provided as a study case of an
application.Comment: To appear in Behavior Research Method
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