Michèle Soria

Marking in combinatorial constructions: generating functions and limiting distributions

There is a wide field of combinatorial constructions, especially in the combinatorial analysis of algorithms, where it is possible to find an explicit generating function y(x) = ?ynxn for the numbers yn of objects of size n and the bivariate generating function y(x, u) = ?ynkxnuk for the numbers ynk of objects of size n where another parameter has value k. Formally this additional parameter is marked in the above combinatorial construction. The aim of this paper is to provide general methods to obtain the asymptotic limiting distribution of this additional parameter in objects of size n.We are especially interested in local limit theorems, which involves estimating the coefficients of powers of generating functions. When y(x) is a function with a logarithmic singularity, we derive uniform approximations for xn]y(x)k for k ? ?n; and as a byproduct, we obtain conditional limiting distributions for the number of trees in random mappings where the number of cycles is given.Production schemas y(x, u) = g(x)F(uw(x)) are also considered: we show how the limiting distribution may be dictated either by g(x), or by F(uw(x)) or should involve both g and F; and give many combinatorial applications.

Introduction for S.I. AofA14

We are delighted to present this special issue of Algorithmica on probabilistic, combinatorial, and asymptotic methods for the analysis of algorithms. Methods that enable precise mathematical analysis of the combinatorial properties of computer programs and data structures have been a focus of a large group of researchers meeting at least annually since the early 1990s. The 2014 meeting in Paris was a special one, marking the occasion of the first Flajolet Lecture, which was delivered by Don Knuth, thus honoring two of the field’s pioneers. This issue consists of seven papers thatwere selected for this issue by the conference program committee. The first two articles, by Drmota and Jin on “An Asymptotic Analysis of Labeled and Unlabeled k-Trees,” and by Krenn and Wagner on “Compositions into Powers of b: Asymptotic Enumeration and Parameters,” continue to expand the frontier of combinatorial structures that can be studied with analytic-combinatoric techniques. The next three articles address classic problems in the analysis of algorithms. In “Analysis of Pivot Sampling in Dual-Pivot Quicksort,” Wild finally develops convincing explanation for the success of Yaroslavskiy’s algorithm in practice; in “On the Cost of Fixed Partial Match Queries in K -d Trees,” Duch, Lau, andMartinez develop a deeper

Obituary: Philippe Flajolet, the Father of Analytic Combinatorics

Philippe Flajolet, mathematician and computer scientist extraordinaire, suddenly passed away on March 22, 2011, at the prime of his career. He is celebrated for opening new lines of research in analysis of algorithms, developing powerful new methods, and solving difficult open problems. His research contributions will have impact for generations, and his approach to research, based on curiosity, a discriminating taste, broad knowledge and interest, intellectual integrity, and a genuine sense of camaraderie, will serve as an inspiration to those who knew him for years to come.

Philippe Flajolet: the father of analytic combinatorics

Philippe Flajolet, mathematician and computer scientist extraordinaire, suddenly passed away on March 22, 2011, at the prime of his career. He is celebrated for opening new lines of research in analysis of algorithms, developing powerful new methods, and solving difficult open problems. His research contributions will have impact for generations, and his approach to research, based on curiosity, a discriminating taste, broad knowledge and interest, intellectual integrity, and a genuine sense of camaraderie, will serve as an inspiration to those who knew him for years to come. The common theme of Flajolet’s extensive and far-reaching body of work is the scientific approach to the study of algorithms, including the development of requisite mathematical and computational tools. During his forty years of research, he contributed nearly 200 publications, with an important proportion of fundamental contributions and representing uncommon breadth and depth. He is best known for fundamental advances in mathematical methods for the analysis of algorithms, and his research also opened new avenues in various domains of applied computer science, including streaming algorithms, communication protocols, database access methods, data mining, symbolic manipulation, text-processing algorithms, and random generation. He exulted in sharing his passion: his papers had more than than a hundred different co-authors and he was a regular presence at scientific meetings all over the world. His research laid the foundation of a subfield of mathematics, now known as analytic combinatorics. His lifework Analytic Combinatorics (Cambridge University Press, 2009, co-authored with R. Sedgewick) is a prodigious achievement that now defines the field and is already recognized as an authoritative reference. Analytic combinatorics is a modern basis for the quantitative study of combinatorial structures (such as words, trees, mappings, and graphs), with applications to probabilistic study of algorithms that are based on these structures. It also strongly influences other scientific domains, such as statistical physics, computational biology, and information theory. With deep historic roots in classical analysis, the basis of the field lies in the work of Knuth, who put the study of algorithms on a firm scientific basis starting in the late 1960s with his classic series of books. Flajolet’s work takes the field forward by introducing original approaches in combinatorics based on two types of methods: symbolic and analytic. The symbolic side is based on the automation of decision procedures in combinatorial enumeration to derive characterizations of generating functions. The analytic side treats those functions as functions in the complex plane and leads to precise characterization of limit distributions. In the last few years, Flajolet was further extending