Dependence between path-length and size in random digital trees

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

Digital Trees and Memoryless Sources: from Arithmetics to Analysis

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.

The variance for partial match retrievals in $k$-dimensional bucket digital trees

International audience The variance of partial match queries in $k$-dimensional tries was investigated in a couple of papers in the mid-nineties, the resulting analysis being long and complicated. In this paper, we are going to re-derive these results with a much easier approach. Moreover, our approach works for $k$-dimensional PATRICIA tries, $k$-dimensional digital search trees and bucket versions as well.

One-sided Variations on Tries: Path Imbalance, Climbing, and Key Sampling

International audience One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.