Skip to Main content Skip to Navigation
Conference papers

Counting Markov Types

Philippe Jacquet 1 Charles Knessl 2 Wojciech Szpankowski 3 
1 HIPERCOM - High performance communication
Inria Paris-Rocquencourt, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so called $\textit{balanced frequency matrix}$ that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length $n$ over an alphabet of size $m$. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special $\textit{linear diophantine equations}$, and also balanced directed multigraphs. For fixed $m$ we prove that the number of Markov types is asymptotically equal to $d(m) \frac{n^{m^{2-m}}}{(m^2-m)!}$, where $d(m)$ is a constant for which we give an integral representation. For $m \to \infty$ we conclude that asymptotically the number of types is equivalent to $\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}$ provided that $m=o(n^{1/4})$ (however, our techniques work for $m=o(\sqrt{n})$). These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method.
Complete list of metadata

Cited literature [19 references]  Display  Hide  Download
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Thursday, August 20, 2015 - 4:32:04 PM
Last modification on : Sunday, June 26, 2022 - 12:03:59 PM
Long-term archiving on: : Wednesday, April 26, 2017 - 9:54:23 AM


Publisher files allowed on an open archive




Philippe Jacquet, Charles Knessl, Wojciech Szpankowski. Counting Markov Types. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.387-400, ⟨10.1109/TIT.2012.2191476⟩. ⟨hal-01185566⟩



Record views


Files downloads