Section: Overall Objectives

Introduction

The primal objective of the project is the field of analysis of algorithms. By this is meant a precise quantification of complexity issues associated to the most fundamental algorithms and data structures of computer science. Departing from traditional approaches that, somewhat artificially, place emphasis on worst-case scenarii, the project focuses on average-case and probabilistic analyses, aiming as often as possible at realistic data models. As such, our research is inspired by the pioneering works of Knuth.

The need to analyze, dimension, and finely optimize algorithms requires an in-depth study of random discrete structures, like words, trees, graphs, and permutations, to name a few. Indeed, a vast majority of the most important algorithms in practice either “make bets” on the likely shape of input data or even base themselves of random choices. In this area we are developing a novel approach based on recent theories of combinatorial analysis together with the view that discrete models connect nicely with complex-analytic and asymptotic methods. The resulting theory has been called “Analytic combinatorics”. Applications of it have been or are currently being worked out in such diverse areas as communication protocols, multidimensional search, data structures for fast retrieval on external storage, data mining applications, the analysis of genomic sequences, and data compression, for instance.

The analytic-combinatorial approach to the basic processes of computer science is very systematic. It appeared early in the history of the project that its development would greatly benefit from the existence of symbolic manipulation systems and computer algebra. This connection has given rise to an original research programme that we are currently carrying out. Some of the directions pursued include automating the manipulation of combinatorial models (counting, generating function equations, random generation), the development of “automatic asymptotics”, and the development of a unified view of the theory of special functions. In particular, the project has developed the Maple library Algolib, that addresses several of these issues.