# Digital search trees and chaos game representation*

Peggy Cénac; Brigitte Chauvin; Stéphane Ginouillac; Nicolas Pouyanne

ESAIM: Probability and Statistics (2009)

- Volume: 13, page 15-37
- ISSN: 1292-8100

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topCénac, Peggy, et al. "Digital search trees and chaos game representation*." ESAIM: Probability and Statistics 13 (2009): 15-37. <http://eudml.org/doc/250658>.

@article{Cénac2009,

abstract = {
In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which one can visualize repetitions of subwords
(seen as suffixes of subsequences). 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, the insertion depth for DST built from independent successive random sequences having the same distribution. Here the successive inserted words are strongly dependent. We give the asymptotic behaviour of the insertion depth and the length of branches for the CGR-tree obtained from the suffixes of a 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.
},

author = {Cénac, Peggy, Chauvin, Brigitte, Ginouillac, Stéphane, Pouyanne, Nicolas},

journal = {ESAIM: Probability and Statistics},

keywords = {Random tree; Digital Search Tree; CGR; lengths of the paths; height; insertion depth; asymptotic growth; strong convergence; random tree; digital search tree; DST; length of branches; asymptotic behaviour},

language = {eng},

month = {2},

pages = {15-37},

publisher = {EDP Sciences},

title = {Digital search trees and chaos game representation*},

url = {http://eudml.org/doc/250658},

volume = {13},

year = {2009},

}

TY - JOUR

AU - Cénac, Peggy

AU - Chauvin, Brigitte

AU - Ginouillac, Stéphane

AU - Pouyanne, Nicolas

TI - Digital search trees and chaos game representation*

JO - ESAIM: Probability and Statistics

DA - 2009/2//

PB - EDP Sciences

VL - 13

SP - 15

EP - 37

AB -
In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which one can visualize repetitions of subwords
(seen as suffixes of subsequences). 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, the insertion depth for DST built from independent successive random sequences having the same distribution. Here the successive inserted words are strongly dependent. We give the asymptotic behaviour of the insertion depth and the length of branches for the CGR-tree obtained from the suffixes of a 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.

LA - eng

KW - Random tree; Digital Search Tree; CGR; lengths of the paths; height; insertion depth; asymptotic growth; strong convergence; random tree; digital search tree; DST; length of branches; asymptotic behaviour

UR - http://eudml.org/doc/250658

ER -

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- * Partially supported by the French Agence Nationale de la Recherche, project SADA ANR-05-BLAN-0372.

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