Section: Scientific Foundations
Keywords : Data Structures, Stochastic Algorithms.
Design and Analysis of Algorithms
The stochastic models of a class of generic algorithms with an underlying tree structure, the splitting algorithms, have a wide range of applications. To classify the massive data sets generated by traffic measurements, these algorithms turn out to be fundamental. Hashing mechanisms such as Bloom filters are currently investigated at the light of these new applications. These algorithms have also been used for now more than 30 years in various areas, among which
Data structures. Fundamental algorithms on data structures are used to sort and search. They are sometimes referred to as divide and conquer algorithms.
Access Protocols. These algorithms are used to give a distributed access to a common communication channel.
Distributed systems. Recently, algorithms to select a subset of a group of identical communicating components like ad hoc networks, sensor networks and more generally mobile networks are using a related approach.
This class of algorithms is fundamental, their range of applications is very large and, moreover, they have a nice underlying mathematical structure. Trees are the main mathematical objects to describe them. The associated stochastic processes can be seen as a discrete version of fragmentation processes which have been recently thoroughly investigated by Bertoin, Pitman and others. They are also related to random recursive decompositions of intervals introduced by Mauldin and Williams and their developments in fractal geometry by Falconer, Lapidus, etc...
A very large subset of the literature has been devoted to the static case analysis, mainly because of its applications in theoretical computer science. In the dynamic case, i.e. when the shape of the tree changes according to some random events, little work has been done for this class of algorithms. Their analysis has been, for the moment, mainly achieved by using analytical methods with functional transforms, complex analysis techniques and inversions theorems. Curiously, despite of the intuitive underlying stochastic structures, probabilistic analyses of these algorithms are quite scarce (see Devroye for example).
Static case . Generalize and simplify the results currently proved by using analytic tools. Prove limit theorems for distributions instead of averages as it is currently the case.
Dynamic case . Study renormalization techniques to analyze tree algorithms under heavy traffic. The understanding of the fundamental features of these algorithms with a traffic of requests is a major issue in this domain. Because of the quite complex technical framework, the partial results obtained up to now with analytical tools hide, in some way, the general phenomena.