Cyclic Complexity of Words

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$Comment: To appear in Journal of Combinatorial Theory, Series

Horn Renamability and Hypergraphs

Satisfiability testing in the context of directed hypergraphs is discussed. A characterization of Horn-renamable formulae is given and a subclass of SAT that belongs to $QTR{cal}{P}$ is described. An algorithm for Horn renaming with linear time complexity is presented

Randomized loose renaming in O(log log n) time

International audienceRenaming is a classic distributed coordination task in which a set of processes must pick distinct identifiers from a small namespace. In this paper, we consider the time complexity of this problem when the namespace is linear in the number of participants, a variant known as loose renaming. We give a non-adaptive algorithm with $O( \log \log n )$ (individual) step complexity, where $n$ is a known upper bound on contention, and an adaptive algorithm with step complexity $O( (\log \log k)^2 )$, where $k$ is the actual contention in the execution. We also present a variant of the adaptive algorithm which requires $O( k \log \log k )$ \emph{total} process steps. All upper bounds hold with high probability against a strong adaptive adversary. We complement the algorithms with an $\Omega( \log \log n )$ expected time lower bound on the complexity of randomized renaming using test-and-set operations and linear space. The result is based on a new coupling technique, and is the first to apply to non-adaptive randomized renaming. Since our algorithms use $O(n)$ test-and-set objects, our results provide matching bounds on the cost of loose renaming in this setting