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Mathematics and
Computer Science III
Algorithms, Trees,
Combinatorics and
Probabilities
Michael Drmota
Philippe Flajolet
Daniele Gardy
Bernhard Gittenberger
Editors
Springer Basel AG
Editors:
Michael Drmota Daniele Gardy
Vienna University of Technology Universite de Versailles-St-Quentin
Institute of Discrete Mathematics PRISM
and Geometry Bâtiment Descartes
Wiedner Hauptstrasse 8-1 O 45 avenue des Etats-Unis
1040Wien 78035 Versailles Cedex
Austria France
e-mail: [email protected] e-mail: [email protected]
Philippe Flajolet Bernhard Gittenberger
INRIA Rocquencourt Vienna University of Technology
78153 Le Chesnay Institute of Discrete Mathematics
France and Geometry
e-mail: [email protected] Wiedner Hauptstrasse 8-1 O
1040 Wien
Austria
e-mail: [email protected]
2000 Mathematical Subject Classification 05-XX, 60C05, 60Gxx, 68P30, 68025, 68Rxx,
68W20, 68W40, 90815
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ISBN 978-3-0348-9620-7 ISBN 978-3-0348-7915-6 (eBook)
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Foreword
These are the Proceedings of the International Colloquium of Mathematics
and Computer Science held at the Vienna University of Technology, September
13-17, 2004. This colloquium is the third one in a now regularly established series
following the first two venues in September 2000 and September 2002 in Ver
sailles. The present issue is centered around Combinatorics and Random Struc
tures, Graph Theory, Analysis of Algorithms, Trees, Probability, Combinatorial
Stochastic Processes, and Applications. It contains invited papers, contributed
papers (lectures) and short communications (posters).
The contributions have been carefully reviewed for their scientific quality and
originality by the Scientific Committee chaired by Michael Drmota (Vienna Uni
versity of Technology, Austria) and composed of Brigitte Chauvin (Universite de
Versailles, France), Luc Devroye (McGill University, Canada), Daniele Gardy (Uni
versite de Versailles, France), Philippe Flajolet (INRIA Rocquencourt, France),
Michal Karonski (Adam Mickiewicz University, Poland), Abdelkader Mokkadem
(Universite de Versailles, France), Helmut Prodinger (University of Witwatersrand,
South Africa), J. Michael Steele (University of Pennsylvania, Philadelphia, USA),
Brigitte Vallee (Universite de Caen, France). We thank them and all anonymous
referees for their impressive work.
We also thank the invited speakers: Jean Bertoin (Universite Paris VI,
France), Mireille Bousquet-Melou (Universite Bordeaux 1, France), Hsien-Kuei
Hwang (Academia Sinica, Taiwan), Svante Janson (Uppsala University, Sweden),
Christian Krattenthaler (Universite Lyon, France), Jean-Franc;ois Marckert (Uni
versite de Versailles, France), Boris Pittel (The Ohio State University, USA), Si
mon Tavare (University of Southern California, USA), the authors of submitted
papers and posters, and the participants for their contribution to the success of
the conference.
Finally, we express our acknowledgements to the Institute of Discrete Math
ematics and Geometry, the Vienna University of Technology, the Federal Ministry
for Education, Science, and Culture, the City of Vienna, the Austrian Research
Society (OFG), the Austrian Mathematical Society (OMG), the Goedel-Society,
and the Bank Austria-Creditanstalt for providing generous financial and material
support.
The Organizing Committee
Bernhard Gittenberger
Thomas Klausner
Alois Panholzer
Preface
These colloquium proceedings address problems at the interface between
Mathematics and Computer Science, with special emphasis on discrete probabilis
tic models and their relation to algorithms. Combinatorial and probabilistic prop
erties of random graphs random trees, combinatorial stochastic processes (such
as random walks) as well as branching processes and related topics in probability
are central. Applications are to be found in the analysis of algorithms and data
structures, the major application field, but also in statistical theory, information
theory, and mathematical logic. This colloquium is the third one in a now regularly
established series, following the first two venues in September 2000 and September
2002 in Versailles. The book features a collection of original refereed contributions:
contributed papers (lectures) and short communications (posters), supplemented
by more detailed articles written by invited speakers (and coauthors): Jean Bertoin
(and Christina Goldschmidt), Svante Janson, and Boris Pittel (and Alan Frieze).
During the final preparation of this volume we received the sad news that
Rainer Kemp (from Frankfurt, Germany) has passed away. Rainer Kemp was one of
the founding fathers of the Analysis of Algorithms, a main topic of our conference.
His book Fundamentals of the average case analysis of particular algorithms, Wiley,
1984, was one of the first books on this subject and had a considerable influence on
the development of the field. He was organizer of several meetings on this subject
and served the scientific community with many other duties. But first of all we
lost a good friend and colleague.
Combinatorics and Random Structures. The starting point of many studies
of random discrete models is combinatorics, which often provides us with exact
representations in terms of counting generating functions that can also be used for
a probabilistic study. Sylvie Corteel, Guy Louchard, and Robin Pemantle work on
the common distribution of intervals in pairs of permutations. Next Sylvie Cor
teel, Jeremy Lovejoy, and Ae Ja Yee provide generating functions for generalized
Frobenius partitions. Luca Ferrari, Renzo Pinzani, and Simone Rinaldi present
some results on integer partitions. Toufik Mansour considers generating functions
for 321-avoiding permutations. Eugenijus ManstaviCius proves an iterated loga
rithm law for the cycle lengths of a random permutation. Martin Rubey provides
a sufficient condition for transcendence of generating functions of walks on the slit
plane. Michael Schlosser finds some curious q-series expansions, and Klaus Simon
presents relations between the numbers of partitions and the divisor functions.
Graph Theory. Graphs are a basic object in discrete mathematics. They are
widely used in applications, and algorithms on graphs as well as theoretical ques
tions on graphs have been "modern topics" of research in mathematics and com
puter science for several decades. Mindaugas Bloznelis uses Hoeffding decompo
sition to prove asymptotic normality of subgraph count statistics. Robert Cori,
Arnaud Dartois, and Dominique Rossin compute so-called "avalanche polynomi
als" for certain families of graphs. The invited paper by Alan Frieze and Boris
Pittel gives a detailed analysis on perfect matchings in random graphs with pre
scribed minimal degree. Omer Gimenez and Marc Noy provide very tight estimates
Vlll Preface
for the growth constant of labelled planar graphs. Finally, Stavros D. Nikolopou
los and Charis Papadopoulos present an algorithm for determining the number of
spanning trees in P4-reducible graphs.
Analysis of Algorithms. This field was created by Donald E. Knuth and is
concerned with accurate estimates of complexity parameters of algorithms and
aims at predicting the behavior of a given algorithm. Javiera Barrera and Chris
tian Paroissin consider specific search cost in random binary search trees. Monia
Bellalouna, Salma Souissi, and Bernard Y cart analyze probabilistic bin packing
problems. Pawel Hitczenko, Jeremy Johnson, and Hung-Jen Huang consider al
gorithms for computing the Walsh-Hadamard transform. Tiimur Ali Khan and
Ralph Neininger analyze the performance of the randomized algorithm to evalu
ate Boolean decision trees proposed by Srnir, in particular they consider the worst
case input and provide limit laws and tail estimated. Next, Shuji Kijima and To
momi Matsui propose a polynomial time perfect sampling algorithm for two-rowed
contingency tables. Conrado Martinez and Xavier Molinero combine two genera
tion algorithms to obtain a new efficient algorithm for the generation of unlabelled
cycles. Finally, Yuriy A. Reznik and Anatoly V. Anisimov suggest the use of tries
for universal data compression.
Trees. Trees are perhaps the most important structure in computer sci
ence. They appear as data structures and are used in various algorithms such as
data compression. David Auber, Jean-Philippe Domenger, Maylis Delest, Philippe
Duchon, and Jean-Marc Fedou present an extension of Strahler numbers to rooted
plane trees. Julien Fayolle analyzes mean size and external path length of a suffix
tree that is related to the LZ'77 data compression algorithm. Eric Fekete considers
two different kinds of external nodes in binary search trees and describes the evo
lution of this process in terms of martingales. The invited paper by Svante Janson
offers an analysis of the number of records in a complete binary tree or equiva
lently the number of random cutting to eliminate a complete binary tree. Interest
ingly the distribution is, after normalization, asymptotically a periodic function in
log n -log log n, where n is the size of the tree. Mehri Javanian and Mohammad Q.
Vahidi-Asl consider multidimensional interval trees. Anne Micheli and Dominique
Rossin describe a specific distance between unlabelled ordered trees, that is based
on deletions and insertions of edges. Katherine Morris determines grand averages
on some parameters in monotonically labelled tree structures. Tatiana Myllari
proves local central limit theorems for the number of vertices of a given outdegree
in a Galton-Watson forest. And finally, Alois Panholzer gives a precise analysis of
the cost distribution for destroying recursive trees in the case of toll functions of
polynomial growth.
Probability. Probabilistic methods get more and more important is the analy
sis of discrete structures: random graphs, random trees, average case analysis of al
gorithms etc. Margaret Archibald addresses the question of the probability that the
maximum in a geometrically distributed sample occurs in the first d positions of a
word. The invited paper by Jean Bertoin and Christina Goldschmidt describes the
duality between a fragmentation associated to certain Dirichlet distributions and
a natural random coagulation. This gives rise to an application to the genealogy
of Yule processes. Mykola S. Bratiychuck considers semi-Markov walks in queue
ing and risk theory. Amke Caliebe characterizes fixed points of linear stochastic
fixed point equations as mixtures of infinitely divisible distributions. Peter Jagers
and Uwe RosIer describe a systematic approach to find solutions of stochastic
fixed points involving the maximum. Arnold Knopfmacher and Helmut Prodinger
Preface
lX
provide central limit theorems for the number of descents in samples of geomet
ric random variables. Alain Rouault proves a law of large numbers and describes
a new large deviation phenomenon for cascades. Christiane Takacs investigates
partitioning properties of piecewise constant eigenvectors of matrices describing
the mutual positions of points. Vladimir Vatutin and Elena Dyakonova consider
branching processes in random environment, find asymptotics of the survival prob
abilities and prove a Yaglom type limit theorem. Finally, Vladimir Vatutin and
Valentin Topchii study the joint distribution of the number of individuals at the
origin and outside the origin on a continuous time random walk on the integers.
Combinatorial Stochastic Processes. Random walks are the most prominent
representatives of combinatorial stochastic processes. They playa central role in
the interplay between combinatorics and probability. Enrica Duchi and Gilles
Schaeffer consider a model of particles jumping on a row of cells with general
boundary conditions where the stationary distribution is not uniform. Guy Fay
olle and Cyril Furtlehner study stochastic deformations of sample paths of random
walks. Johannes Fehrenbach and Ludger Riischendorf show that a Markov chain
that is naturally defined on the Eulerian orientation of planar graph converges to
uniform distribution. Alexander Gnedin considers regenerative composition struc
tures. Jean Mairesse and Frederic Matheus study transient nearest neighbor ran
dom walks on groups with a finite set of generators and compute various char
acteristics such as the drift and the entropy. Finally, Philippe Marchal gives a
fractal construction of nested, stable regenerative sets and studies the associated
inhomogeneous fragmentation process.
Applications. Random combinatorics interacts with many other areas of sci
ence. Eda Cesaratto and Brigitte Vallee consider numeration schemes, defined in
terms of dynamical systems and determine the Hausdorff dimension of sets of reals
which obey some constraints on their digits. Adriana Climescu -Haulica deals with
large deviation analysis of space-time Trellis codes. David Coupier, Agnes Desol
neux, and Bernard Y cart provide a zero-one law for first order logic on random
images. Nadia Creignou and Herve Daude study threshold phenomena for random
generalized satisfyability problems. Guy Fayolle, Vadim Malyshev, and Serguei
Pirogov introduce new models of energy redistribution in stochastic chemical ki
netics with several molecule types and energy parameters. Laszlo Gyorfi discusses
Chernoff type large deviations of Hellinger distance on partitions. Nadia Lalam
and Christine Jacob address the problem of estimating the offspring mean for a
general class of size-dependent branching processes. Malgorzata and Wlodzimierz
Moczurad deal with the problem of decidability of simple brick codes. And finally,
Joel Ratsaby generalizes Sauer's Lemma to finite VC-dimension classes of binary
valued functions.
Altogether papers assembled in this volume offer snapshots of current re
search. At the same time, they illustrate the numerous ramifications of the the
ory of random discrete structures throughout mathematics and computer science.
Many of them, in particular invited lectures, include carefully crafted surveys of
their field. We thus hope that the book may serve both as a reference text and as a
smooth introduction to many fascinating aspects of this melting pot of continuous
and discrete mathematics.
Michael Drmota
Philippe Flajolet
Daniele Gardy
Bernhard Gittenberger
Contents
PART I. Combinatorics and Random Structures
Common Intervals of Permutations
Sylvie Corteel, Guy Louchard, and Robin Pemantle 3
Overpartitions and Generating Functions for Generalized Frobenius Partitions
Sylvie Corteel, Jeremy Lovejoy, and Ae Ja Vee 15
Enumerative Results on Integer Partitions Using the ECO Method
Luca Ferrari, Renzo Pinzani, and Simone Rinaldi 25
321-Avoiding Permutations and Chebyshev Polynomials
Toufik Mansour 37
Iterated Logarithm Laws and the Cycle Lengths of a Random Permutation
Eugenijus ManstaviCius 39
Transcendence of Generating Functions of Walks on the Slit Plane
Martin Rubey 49
Some Curious Extensions of the Classical Beta Integral Evaluation
Michael Schlosser 59
Divisor Functions and Pentagonal Numbers
Klaus Simon 69
PART II. Graph Theory
On Combinatorial Hoeffding Decomposition and Asymptotic Normality of
Subgraph Count Statistics
Mindaugas Bloznelis 73
Avalanche Polynomials of Some Families of Graphs
Robert Cori, Arnaud Dartois, and Dominique Rossin 81
Perfect Matchings in Random Graphs with Prescribed Minimal Degree
Alan Frieze and Boris Pittel 95
Estimating the Growth Constant of Labelled Planar Graphs
Orner Gimenez and Marc Noy 133
The Number of Spanning Trees in P4-Reducible Graphs
Stavros D. Nikolopoulos and Charis Papadopoulos 141
xu Contents
PART III. Analysis of Algorithms
On the Stationary Search Cost for the Move-to-Root Rule with Random
Weights
Javiera Barrera and Christian Paroissin 147
Average-Case Analysis for the Probabilistic Bin Packing Problem
Monia Bellalouna, Salma Souissi, and Bernard Y cart 149
Distribution of WHT Recurrences
Pawel Hitczenko, Jeremy R. Johnson, and Hung-Jen Huang 161
Probabilistic Analysis for Randomized Game 'free Evaluation
Tamur Ali Khan and Ralph Neininger 163
Polynomial Time Perfect Sampling Algorithm for Two-Rowed Contingency
Tables
Shuji Kijima and Tomomi Matsui 175
An Efficient Generic Algorithm for the Generation of Unlabelled Cycles
Conrado Martinez and Xavier Molinero 187
Using 'fries for Universal Data Compression
Yuriy A. Reznik and Anatoly V. Anisimov 199
PART IV. Trees
New Strahler Numbers for Rooted Plane 'frees
David Auber, Jean-Philippe Domenger, Maylis Delest, Philippe Duchon, and
Jean-Marc Fedou 203
An Average-Case Analysis of Basic Parameters of the Suffix 'free
Julien Fayolle 217
Arms and Feet Nodes Level Polynomial in Binary Search 'frees
Eric Fekete 229
Random Records and Cuttings in Complete Binary 'frees
Svante Janson 241
Multidimensional Interval 'frees
Mehri Javanian and Mohammad Q. Vahidi-Asl 255
Edit Distance between Unlabelled Ordered 'frees
Anne Micheli and Dominique Rossin 257
On Parameters in Monotonically Labelled 'frees
Katherine Morris 261
