124 research outputs found
The complexity of the Multiple Pattern Matching Problem for random strings
We generalise a multiple string pattern matching algorithm, recently proposed
by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary
dictionaries on an alphabet of size . If is the number of words of
length in the dictionary, and , the
complexity rate for the string characters to be read by this algorithm is at
most for some constant
. On the other side, we generalise the classical lower
bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern,
to deal with arbitrary dictionaries, and determine it to be at least
. This proves the optimality of the
algorithm, improving and correcting previous claims.Comment: 25 pages, 4 figure
On the genericity of Whitehead minimality
We show that a finitely generated subgroup of a free group, chosen uniformly
at random, is strictly Whitehead minimal with overwhelming probability.
Whitehead minimality is one of the key elements of the solution of the orbit
problem in free groups. The proofs strongly rely on combinatorial tools,
notably those of analytic combinatorics. The result we prove actually depends
implicitly on the choice of a distribution on finitely generated subgroups, and
we establish it for the two distributions which appear in the literature on
random subgroups
Generic properties of subgroups of free groups and finite presentations
Asymptotic properties of finitely generated subgroups of free groups, and of
finite group presentations, can be considered in several fashions, depending on
the way these objects are represented and on the distribution assumed on these
representations: here we assume that they are represented by tuples of reduced
words (generators of a subgroup) or of cyclically reduced words (relators).
Classical models consider fixed size tuples of words (e.g. the few-generator
model) or exponential size tuples (e.g. Gromov's density model), and they
usually consider that equal length words are equally likely. We generalize both
the few-generator and the density models with probabilistic schemes that also
allow variability in the size of tuples and non-uniform distributions on words
of a given length.Our first results rely on a relatively mild prefix-heaviness
hypothesis on the distributions, which states essentially that the probability
of a word decreases exponentially fast as its length grows. Under this
hypothesis, we generalize several classical results: exponentially generically
a randomly chosen tuple is a basis of the subgroup it generates, this subgroup
is malnormal and the tuple satisfies a small cancellation property, even for
exponential size tuples. In the special case of the uniform distribution on
words of a given length, we give a phase transition theorem for the central
tree property, a combinatorial property closely linked to the fact that a tuple
freely generates a subgroup. We then further refine our results when the
distribution is specified by a Markovian scheme, and in particular we give a
phase transition theorem which generalizes the classical results on the
densities up to which a tuple of cyclically reduced words chosen uniformly at
random exponentially generically satisfies a small cancellation property, and
beyond which it presents a trivial group
Optimal prefix codes for pairs of geometrically-distributed random variables
Optimal prefix codes are studied for pairs of independent, integer-valued
symbols emitted by a source with a geometric probability distribution of
parameter , . By encoding pairs of symbols, it is possible to
reduce the redundancy penalty of symbol-by-symbol encoding, while preserving
the simplicity of the encoding and decoding procedures typical of Golomb codes
and their variants. It is shown that optimal codes for these so-called
two-dimensional geometric distributions are \emph{singular}, in the sense that
a prefix code that is optimal for one value of the parameter cannot be
optimal for any other value of . This is in sharp contrast to the
one-dimensional case, where codes are optimal for positive-length intervals of
the parameter . Thus, in the two-dimensional case, it is infeasible to give
a compact characterization of optimal codes for all values of the parameter
, as was done in the one-dimensional case. Instead, optimal codes are
characterized for a discrete sequence of values of that provide good
coverage of the unit interval. Specifically, optimal prefix codes are described
for (), covering the range , and
(), covering the range . The described codes produce the expected
reduction in redundancy with respect to the one-dimensional case, while
maintaining low complexity coding operations.Comment: To appear in IEEE Transactions on Information Theor
Deciding the finiteness of the number of simple permutations contained in a wreath-closed class is polynomial
We present an algorithm running in time O(n ln n) which decides if a
wreath-closed permutation class Av(B) given by its finite basis B contains a
finite number of simple permutations. The method we use is based on an article
of Brignall, Ruskuc and Vatter which presents a decision procedure (of high
complexity) for solving this question, without the assumption that Av(B) is
wreath-closed. Using combinatorial, algorithmic and language theoretic
arguments together with one of our previous results on pin-permutations, we are
able to transform the problem into a co-finiteness problem in a complete
deterministic automaton
Enumeration of Pin-Permutations
39 pagesInternational audienceIn this paper, we study the class of pin-permutations, that is to say of permutations having a pin representation. This class has been recently introduced in an article of Brignall, Huczinska and Vatter, where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of the substitution decomposition trees of pin-permutations, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured, the rationality of this generating function. Moreover, we show that the basis of the pin-permutation class is infinite
Automates, énumération et algorithmes
Ces travaux s'inscrivent dans le cadre général de la théorie des automates, de la combinatoire des mots, de la combinatoire énumérative et de l'algorithmique. Ils ont en commun de traiter des automates et des langages réguliers, de problÚmes d'énumération et de présenter des résultats constructifs, souvent explicitement sous forme d'algorithmes. Les domaines dont sont issus les problÚmes abordés sont assez variés. Ce texte est compose de trois parties consacrées aux codes préfixes, à certaines séquences lexicographiques et à l'énumération d'automates
Combinatorial specification of permutation classes
This article presents a methodology that automatically derives a
combinatorial specification for the permutation class C = Av(B), given its
basis B of excluded patterns and the set of simple permutations in C, when
these sets are both finite. This is achieved considering both pattern avoidance
and pattern containment constraints in permutations.The obtained specification
yields a system of equations satisfied by the generating function of C, this
system being always positiveand algebraic. It also yields a uniform random
sampler of permutations in C. The method presentedis fully algorithmic
On the genericity of Whitehead minimality
International audienceWe show that a finitely generated subgroup of a free group, chosen uniformly at random, is strictly Whitehead minimal with overwhelming probability. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on combinatorial tools, notably those of analytic combinatorics. The result we prove actually depends implicitly on the choice of a distribution on finitely generated subgroups, and we establish it for the two distributions which appear in the literature on random subgroups
- âŠ