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A First Investigation of Sturmian Trees [chapter]

Jean Berstel, Luc Boasson, Olivier Carton, Isabelle Fagnot
STACS 2007  
We consider Sturmian trees as a natural generalization of Sturmian words. A Sturmian tree is a tree having n+1 distinct subtrees of height n for each n.  ...  We give various examples of Sturmian trees, and we characterize one family of Sturmian trees by means of a structural property of their automata.  ...  The first four words ε, 01, 0101 and 0011 of L correspond to the four occurrences of the letter a as label on the top of the tree.  ... 
doi:10.1007/978-3-540-70918-3_7 dblp:conf/stacs/BerstelBCF07 fatcat:ougbsfjqvnbirpbj6ocd4r57du

Circular sturmian words and Hopcroft's algorithm

G. Castiglione, A. Restivo, M. Sciortino
2009 Theoretical Computer Science  
We prove that the shape of the reduction tree of a circular sturmian word (x) coincides with that of the derivation tree T (A x ) of the automaton A x .  ...  In order to analyze some extremal cases of Hopcroft's algorithm, we investigate the relationships between the combinatorial properties of a circular sturmian word (x) and the run of the algorithm on the  ...  In the first part of the paper we investigate some combinatorial properties of circular sturmian words.  ... 
doi:10.1016/j.tcs.2009.07.018 fatcat:7x3khicglrcjnl3isvyh3avmvq

Page 4406 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
First, we investigate structural properties of the executions of such a dynamic process graph Y.  ...  Sci. 255 (2001), no. 1-2, 539-553; MR 2002c:68073] begun a study of a slightly more general class of infinite words that we call epi-Sturmian.  ... 

Substitution invariant Sturmian words and binary trees [article]

Michel Dekking
2017 arXiv   pre-print
We show that homogeneous substitution invariant Sturmian sequences s_α,α can be indexed by two binary trees, associated directly to Johannes Kepler's tree of harmonic fractions from 1619.  ...  We take a global view at substitution invariant Sturmian sequences.  ...  Below we display the first three levels of the tree of Sturm numbers α with α < 1 2 associated to the morphisms of T ϕ .  ... 
arXiv:1705.08607v1 fatcat:uz3jp7m4jrgbhcpigv7lxk6fny

Infinite labeled trees: From rational to Sturmian trees

Nicolas Gast, Bruno Gaujal
2010 Theoretical Computer Science  
The first part of the paper is dedicated to the study of general unordered infinite trees with binary labels.  ...  Section 3 is dedicated to the study of the rational trees. The second part of the paper investigates balanced unordered trees and their properties.  ...  Density of a balanced tree Before beginning the full investigation of balanced trees, we start with a rather straightforward property: a balanced tree has a density.  ... 
doi:10.1016/j.tcs.2009.12.009 fatcat:22hgfcxbpngn5mttfh7pstokhq

Distributing Labels on Infinite Trees [article]

Nicolas Gast, Bruno Gaujal
2008 arXiv   pre-print
In this paper, we consider the problem of generalizing Sturmian words to trees. The problem is to evenly distribute labels 0 and 1 over infinite trees.  ...  Therefore they bring the hope that extremal scheduling properties of Sturmian words can be extended to such trees, as least partially. Such possible extensions are illustrated by one such example.  ...  of the paper investigates balanced unordered trees and their properties.  ... 
arXiv:0809.1989v1 fatcat:qj4mauinjba4zn6hikezdrwb4y

Page 5108 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
Our main result shows that any spanning tree defined by the edges corresponding to a first visit of a vertex 68 COMPUTER SCIENCE 5108 appears with a probability proportional to its weight, which is the  ...  the edges, are investigated.  ... 

Lyndon factorization of sturmian words

Guy Melançon
2000 Discrete Mathematics  
We also give a short proof of a result by Berstel and de Luca (Sturmian words, Lyndon words and trees, Theoret. Comput.  ...  We express any general characteristic sturmian word as a unique inÿnite non-increasing product of Lyndon words. Using this identity, we give a new !-division for characteristic sturmian words.  ...  First, we prove that the factorization of s gives an !-division for it.  ... 
doi:10.1016/s0012-365x(99)00123-5 fatcat:rumv6a44l5ehrghysv4taw7wya

Page 8132 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
The performance of the estimates is measured by the robust L, error. The convergence and rates of the estimates are investigated.”  ...  In the case of infinite words they prove that knowledge of the cardinality of M, is enough to characterize the set of factors of x in case x is a Sturmian word (i.e. a word which has exactly n+1 different  ... 

Page 3884 of Mathematical Reviews Vol. , Issue 98F [page]

1998 Mathematical Reviews  
Kapoor and Ramesh first proposed an O(N + V + E)-time algorithm by adopting such a ‘compact’ out- put, where N is the number of spanning trees.  ...  Shioura and A. Tamura [J. Oper. Res. Soc. Japan 38 (1995), no. 3, 331-344; MR 96f:05094]. They find a new spanning tree by exchanging one edge of a cur- rent one.  ... 

Quasi-Sturmian colorings on regular trees [article]

Dong Han Kim, Seul Bee Lee, Seonhee Lim, Deokwon Sim
2018 arXiv   pre-print
Quasi-Sturmian words, which are infinite words with factor complexity eventually n+c share many properties with Sturmian words. In this paper, we study the quasi-Sturmian colorings on regular trees.  ...  There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.  ...  Factor complexity and Sturmian words have been generalized to Sturmian colorings of trees by the first and third authors [6] .  ... 
arXiv:1808.05400v1 fatcat:l66zwermhfhtzgpf3he27zxvrm

Page 7292 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
98k:68139 edge problem with respect to a minimum spanning tree is to find a set S (C E) of k edges in G whose removal results in the greatest increase in the weight of the minimum spanning tree in the  ...  of their factors, or with the well-known Sturmian words).  ... 

Page 6292 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
A Sturmian sequence is an infinite binary word which contains exactly n +1 different factors of length » for each nonnegative integer ». A Sturmian word is any finite factor of a Sturmian sequence.  ...  The authors prove that if U is an unbordered Sturmian word then every Duval extension W of U is of the form W = U"’ with ui . . : a $ r = —~.  ... 

Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces [article]

Jean V. Bellissard, Antoine Julien
2013 arXiv   pre-print
An ultrametric Cantor set can be seen as the boundary of a rooted weighted tree called the Michon tree. The notion of Assouad dimension is re-interpreted as seen on the Michon tree.  ...  It is applied to answer the embedding question for some spaces which can be seen naturally as boundary of trees: linearly repetitive subshifts, Sturmian subshifts, and the boundary of Galton--Watson trees  ...  B. thanks A. Grigor'yan and his group at the CRC701, University of Bielefeld for giving him the opportunity to present this result prior to publication. A. J. thanks J.  ... 
arXiv:1202.4330v2 fatcat:r7nclmpnjfa45dih24mbwlzojy

Rigidity and Substitutive Dendric Words

V. Berthé, F. Dolce, F. Durand, J. Leroy, D. Perrin
2018 International Journal of Foundations of Computer Science  
We investigate here the properties of substitutive tree words and prove some rigidity properties, that is, algebraic properties on the set of substitutions that fix a tree word.  ...  This class of words includes classical families of words such as Sturmian and episturmian words, codings of interval exchanges or else Arnoux-Rauzy words.  ...  We investigate here the properties of substitutive tree words and prove some rigidity properties.  ... 
doi:10.1142/s0129054118420017 fatcat:ljboyxp46ndfhgfvq265kt6sr4
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