Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1404.4189

Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincaré Algorithm

Valérie Berthé·Sébastien Labbé
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Arnoux-Rauzy-Poincaré multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincaré algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in subtracting the sum of the two smallest entries to the largest if possible (Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the median and the median to the largest (the Poincaré step), and by performing when possible Arnoux-Rauzy steps in priority. After renormalization it provides a piecewise fractional map of the standard $2$-simplex. We study here the factor complexity of its associated symbolic dynamical system, defined as an $S$-adic system. It is made of infinite words generated by the composition of sequences of finitely many substitutions, together with some restrictions concerning the allowed sequences of substitutions expressed in terms of a regular language. Here, the substitutions are provided by the matrices of the linear version of the algorithm. We give an upper bound for the linear growth of the factor complexity. We then deduce the convergence of the associated algorithm by unique ergodicity.

Explore related subjects

cs.DMmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Valérie Berthé, Sébastien Labbé. 2014-04-16. Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincaré Algorithm. https://doi.org/10.1016/j.aam.2014.11.001

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Three Hardness Results for Graph Similarity Problems

Notions of graph similarity provide alternative perspective on the graph isomorphism problem and vice-versa. In this paper, we consider measures of similarity arising from mismatch norms as studied in Gervens and Grohe: the edit distance $δ_{\mathcal{E}}$, and the metrics arising from $\ell_p$-operator norms, which we denote by $δ_p$ and $δ_{|p|}$. We address the following question: can these measures of similarity be used to design polynomial-time approximation algorithms for graph isomorphism? We show that computing an optimal value of $δ_{\mathcal{E}}$ is \NP-hard on pairs of graphs with the same number of edges. In addition, we show that computing optimal values of $δ_p$ and $δ_{|p|}$ is \NP-hard even on pairs of $1$-planar graphs with the same degree sequence and bounded degree. These two results improve on previous known ones, which did not examine the restricted case where the pairs of graphs are required to have the same number of edges. Finally, we study similarity problems on strongly regular graphs and prove some near optimal inequalities with interesting consequences on the computational complexity of graph and group isomorphism.

cs.DM↗
arXiv2026

Counting and Covering in Nearest-Neighbour Representations of Boolean Functions

We study the number of prototypes needed to represent Boolean functions by nearest-neighbour classification. There are two distinct settings: the prototypes may be arbitrary points of Euclidean space, or they may themselves be required to lie in the Boolean cube. For unrestricted prototypes, we strengthen a known lower bound for almost all Boolean functions. The bound applies simultaneously to nearest-neighbour voting rules with any number of voting neighbours, and substantially narrows the gap with the known general upper bound. We obtain a VC-dimension bound for classes with a bounded number of prototypes, and show that it is sharp in order in dimensions at least four. We then study Boolean prototypes, beginning with symmetric threshold functions. A connection with covering designs expresses the minimum number of prototypes at every threshold level exactly in terms of a covering number, and leads to further exact results for related monotone functions, including disjunctive extensions and a characterisation of when a representation with a single negative prototype is possible. For a uniformly random Boolean function, the Boolean nearest-neighbour complexity, as a proportion of the cube, is asymptotically close either to one half or to one, with explicit limiting probabilities. In particular, almost every Boolean function requires at least approximately half as many prototypes as there are points in the cube, and one half is the largest proportion for which such a lower bound holds. Finally, we consider arbitrary symmetric Boolean functions. Their Boolean nearest-neighbour complexity is closely approximated by a weighted vertex-cover problem on paths. As a consequence, a uniformly random symmetric function typically requires prototypes amounting to $11/20$ of the cube. This is much larger than the upper bounds known when the prototypes are allowed to lie anywhere in Euclidean space.

cs.DM↗
arXiv2026

Super-linear Lower Bounds for CSP Non-Redundancy via Shrinking Instances

We say that an instance of a constraint satisfaction problem (CSP) is non-redundant if the satisfaction of each clause cannot be implied by the satisfaction of the other clauses in the instance. The non-redundancy (NRD) of a CSP is the maximal number of clauses a non-redundant instance can have for a given number of variables. NRD is closely tied to the behavior of CSPs in various computational models including their sparsification, kernelization, and streaming complexity. A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD. Recent works by Carbonnel [CP 2022], Khanna, Putterman and Sudan [STOC 2025], Brakensiek and Guruswami [STOC 2025] and Brakensiek, Guruswami, Jansen, Lagerkvist, and Wahlström [2025] have introduced various forms of gadget reductions between CSPs to relate their non-redundancy. The primary contribution of this work is to recontextualize many of these gadget reductions in a framework which we call hypergraph projections. By studying a quantity we call the shrinking factor of these hypergraph projections, we can more precisely predict when a gadget reduction between predicates can yield a super-linear NRD lower bound, greatly improving on the analysis of previous works. To illustrate the power of our framework, we identify some concrete CSP predicates whose non-redundancy is at the cusp of our understanding and show how our methods give lower bounds that could not have been achieved with previous methods. We also demonstrate how these gadget reductions can be automatically deduced using SAT solvers, thereby opening up novel computational avenues for discovering further relationships between the non-redundancy of various CSPs.

cs.DM↗