Digital search trees and chaos game representation

In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which on can visualize repetitions of subwords. The CGR-tree turns a sequence of letters into a digital search tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height and the insertion depth for DST built from i.i.d. successive sequences. Here, the successive inserted wors are strongly dependent. We give the asymptotic behaviour of the insertion depth and of the length of branches for the CGR-tree obtained from the suffixes of reversed i.i.d. or Markovian sequence. This behaviour turns out to be at first order the same one as in the case of independent words. As a by-product, asymptotic results on the length of longest runs in a Markovian sequence are obtained

Variable length Markov chains and dynamical sources

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the ``comb'' and the ``bamboo blossom'', we find a necessary and sufficient condition for the existence and the unicity of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences.Comment: 45 pages, 15 figure

Digital search trees and chaos game representation

Version préliminaire (2006) d'un travail publié sous forme définitive (2009).International audienceIn this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which on can visualize repetitions of subwords. The CGR-tree turns a sequence of letters into a digital search tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height and the insertion depth for DST built from i.i.d. successive sequences. Here, the successive inserted wors are strongly dependent. We give the asymptotic behaviour of the insertion depth and of the length of branches for the CGR-tree obtained from the suffixes of reversed i.i.d. or Markovian sequence. This behaviour turns out to be at first order the same one as in the case of independent words. As a by-product, asymptotic results on the length of longest runs in a Markovian sequence are obtained

Characterization of stationary probability measures for Variable Length Markov Chains

By introducing a key combinatorial structure for words produced by a Variable Length Markov Chain (VLMC), the longest internal suffix, precise characterizations of existence and uniqueness of a stationary probability measure for a VLMC chain are given. These characterizations turn into necessary and sufficient conditions for VLMC associated to a subclass of probabilised context trees: the shift-stable context trees. As a by-product, we prove that a VLMC chain whose stabilized context tree is again a context tree has at most one stationary probability measure. MSC 2010: 60J05, 60C05, 60G10

Characterization of stationary probability measures for Variable Length Markov Chains

32 pagesBy introducing a key combinatorial structure for words produced by a Variable Length Markov Chain (VLMC), the longest internal suffix, precise characterizations of existence and uniqueness of a stationary probability measure for a VLMC chain are given. These characterizations turn into necessary and sufficient conditions for VLMC associated to a subclass of probabilised context trees: the shift-stable context trees. As a by-product, we prove that a VLMC chain whose stabilized context tree is again a context tree has at most one stationary probability measure

Context trees, variable length Markov chains and dynamical sources.

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the "comb" and the "bamboo blossom", we find a necessary and sufficient condition for the existence and the uniqueness of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences

Uncommon suffix tries

International audienc

Variable length Markov chains and dynamical sources

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the ``comb'' and the ``bamboo blossom'', we find a necessary and sufficient condition for the existence and the unicity of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences

Uncommon Suffix Tries

Common assumptions on the source producing the words inserted in a suffix trie with $n$ leaves lead to a $\log n$ height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of $n$ and another one whose saturation level is negligible with respect to $\log n$. Both are built from VLMC (Variable Length Markov Chain) probabilistic sources; they are easily extended to families of sources having the same properties. The first example corresponds to a ''logarithmic infinite comb'' and enjoys a non uniform polynomial mixing. The second one corresponds to a ''factorial infinite comb'' for which mixing is uniform and exponential