Digital Trees and Memoryless Sources: from Arithmetics to Analysis

Abstract

International audience Digital trees, also known as $\textit{"tries''}$, are fundamental to a number of algorithmic schemes, including radix-based searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the $\textit{memoryless source}$, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters' expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous $\textit{Dirichlet series}$. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific $\textit{arithmetic properties}$, especially irrationality measures, of the sources under consideration.

Context Trees, Variable Length Markov Chains and Dynamical Sources

“…and Formulae (11), ( 13) and (8). Furthermore, π is trivial if and only if q 1 (0) = 0, in which case it is defined by π(1 ∞ ) = 1.…”

“…In this case, the source is memoryless: all letters are drawn independently with the same distribution. The Dirichlet series of such sources have been extensively studied in Flajolet et al [8] in the realm of asymptotics of average parameters of a trie.…”

Average Size of a Suffix Tree for Markov Sources

“…Remark We recall that a set of real numbers are commensurable (also known as "rationally related") when their ratios are rational numbers. We observe that if for all (a, b) ∈ A 2 , the α abc are commensurable for one c ∈ A, then α abc are commensurable for all values of c.Furthermore in the aperiodic case the o(n) term can have a growth rate arbitrary close to order n, depending on source settings as shown in [7] in the memoryless case.…”

Asymptotic Normality for the Size of Graph Tries built from M-ary Tree Labelings

“…A great deal about these sets is known; see the deep study of Flajolet et al [3] for a slightly restricted setting which, however, carries over to the solution set of P (s), too. First, the structure of the set S ρ depends on a property of the ratios log p i / log p j .…”

“…For instance, the roots are all simple, are uniformly separated, 1/P (z) is bounded provided that z stays uniformly far away from the roots, etc. ; see [3] for more properties.…”