26,492 research outputs found
Finding Pseudo-repetitions
Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word
Promised streaming algorithms and finding pseudo-repetitions
As the size of data available for processing increases, new models of
computation are needed. This motivates the study of data streams, which are sequences of
information for which each element can be read only after the previous one. In
this work we study two particular types of streaming variants: promised graph streaming algorithms and
combinatorial queries on large words. We give an &omega(n) lower
bound for working memory, where n is the number of vertices of the graph, for a variety
of problems for which the graphs are promised to be forests. The crux
of the proofs is based on reductions from the field of communication complexity.
Finally, we give an upper bound for two problems related to finding
pseudo-repetitions on words via anti-/morphisms, for which we also propose streaming versions
Testing Generalised Freeness of Words
Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding efficiently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variable x and its image under f, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect efficiently a factor of the form xx^Rxxx^R, or any other pattern of such type. We also address the problem of testing efficiently, in the same setting, whether the word w contains an arbitrary pseudo-repetition of a given exponent
A Minimal Periods Algorithm with Applications
Kosaraju in ``Computation of squares in a string'' briefly described a
linear-time algorithm for computing the minimal squares starting at each
position in a word. Using the same construction of suffix trees, we generalize
his result and describe in detail how to compute in O(k|w|)-time the minimal
k-th power, with period of length larger than s, starting at each position in a
word w for arbitrary exponent and integer . We provide the
complete proof of correctness of the algorithm, which is somehow not completely
clear in Kosaraju's original paper. The algorithm can be used as a sub-routine
to detect certain types of pseudo-patterns in words, which is our original
intention to study the generalization.Comment: 14 page
Computationally efficient algorithms for the two-dimensional Kolmogorov-Smirnov test
Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2^d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n^2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of
Omega(n^2.lg(n)), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test
A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs
We study implications of unitarity for pseudo-orbit expansions of the
spectral determinants of quantum maps and quantum graphs. In particular, we
advocate to group pseudo-orbits into sub-determinants. We show explicitly that
the cancellation of long orbits is elegantly described on this level and that
unitarity can be built in using a simple sub-determinant identity which has a
non-trivial interpretation in terms of pseudo-orbits. This identity yields much
more detailed relations between pseudo orbits of different length than known
previously. We reformulate Newton identities and the spectral density in terms
of sub-determinant expansions and point out the implications of the
sub-determinant identity for these expressions. We analyse furthermore the
effect of the identity on spectral correlation functions such as the
auto-correlation and parametric cross correlation functions of the spectral
determinant and the spectral form factor.Comment: 25 pages, one figur
Detecting One-variable Patterns
Given a pattern such that
, where is a
variable and its reversal, and
are strings that contain no variables, we describe an
algorithm that constructs in time a compact representation of all
instances of in an input string of length over a polynomially bounded
integer alphabet, so that one can report those instances in time.Comment: 16 pages (+13 pages of Appendix), 4 figures, accepted to SPIRE 201
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