Program

The electronic journal Discrete Mathematics and Theoretical Computer Science defines the field of Analysis of Algorithms as follows: Analysis of algorithms is concerned with accurate estimates of complexity parameters of algorithms and aims at predicting the behavior of a given algorithm run in a given environment. It develops general methods for obtaining closed-form formulae, asymptotic estimates, and probability distributions for combinatorial or probabilistic quantities, that are of interest in the optimization of algorithms. Interest is also placed on the methods themselves, whether combinatorial, probabilistic, or analytic. Combinatorial and statistical properties of discrete structures (strings, trees, tries, dags, graphs, and so on) as well as mathematical objects (e.g., continued fractions, polynomials, operators) that are relevant to the design of efficient algorithms are investigated. Since its inception in 1963, the field has undergone substantial changes. We see now the emergence of combinatorial and asymptotic methods that allow the classification of data structures into broad categories that are amenable to a unified treatment. In recent years, probabilistic methods, which have been so successful in the study of random graphs and hard combinatorial optimization problems, have also been shown to play an important role in this field. These developments have two important consequences for the analysis of algorithms: it becomes possible to predict average behavior under more general probabilistic models than before; at the same time it becomes possible to analyze algorithms that are more structurally complex than before. To achieve these goals, the analysis of algorithms draws on a number of branches of mathematics: combinatorics, probability theory, graph theory, real and complex analysis, number theory, and occasionally algebra, geometry, operations research, and others. Information theory appears to be an area of strong synergy with analysis of algorithms. This synergy has already yielded very interesting results, such as the characterization of structural properties of the Lempel-Ziv parsing tree and a precise asymptotic characterization of the redundancy of the Lempel-Ziv data compression scheme. Thus, techniques from analysis of algorithms have given new insights into the structure of the algorithms used in information theory, as well as their information-theoretic performance. The first half of this summer graduate program will focus on analytic combinatorics and algorithms, while information theory applications will be studied in the second. Lecturers will include P. Flajolet, G. Seroussi, W. Szpankowski, and M. Weinberger. This summer graduate program will be followed by a one-week workshop on analysis of algorithms.