Algebraic Structures and Operator Calculus
Mathematics and Its Applications
Managing Editor: M . HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Also of interest: Algebraic Structures and Operator Calculus . Volume I.- Representations and Probability
Theory, by P. Feinsilver and R . Schott, 1993, x + 224 pp., ISBN 0-7923-2116-2, MAIA 241 .
Volume 292
Algebraic Structures and Operator Calculus Volume II : Special Functions and Computer Science
by Philip Feinsilver Department of Mathematics, Southern Illinois University, Carbondale, Illinois, U.S.A .
and Rene Schott CRIN, Universite; de Nancy 1, Vandoeuvre-les-Nancy, France
1
0
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
A C .I .P. Catalogue record for this book is available from the Library of Congress .
ISBN 0-7923-2921-X
Published by Kluwer Academic Publishers, P.O . Box 17, 3300 AA Dordrecht, The Netherlands . Kluwer Academic Publishers incorporates the publishing programmes of D . Reidel, Martinus Nijhoff, Dr W . Junk and MT? Press. Sold and distributed in the U .S .A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A . In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands .
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All Rights Reserved ® 1994 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner . Printed in the Netherlands
To our parents In memoriam : Leo and Cecilia, Joseph and Anne
Table of Contents
Preface
ix
Introduction I . General remarks II . Some notations III . Orthogonal polynomials and continued fractions IV . Bessel functions, Lommel polynomials, theta functions V . Some analytic techniques VI . Polynomials : reference information
1 2 3 5 7 9
Chapter 1 Basic Data Structures I . Basic data structures II . Dynamic data structures
14 18
Chapter 2 Data Structures and Orthogonal Polynomials I . Dynamic data structures : Knuth's model II. Mutual exclusion III. Elements of duality theory
24 49 52
Chapter 3 Applications of Bessel Functions and Lommel Polynomials I. Analysis of the symbol table in Knuth's model II . Analysis of some concurrent algorithms
54 65
Chapter 4 Fourier Transform on Finite Groups and Related Transforms I . Basic definitions and properties 76 II . Fourier transform on finite abelian groups 89 III . Krawtchouk polynomials and finite probability theory 91 IV . Appell states 99 V . Orthogonal polynomial expansions via Fourier transform 104 Chapter 5 YoungTableaux and Combinatorial Enumeration in Parallel Processing I . Representations of the symmetric group 112 II . Clustering distributions 122 III . Inversion of circulants 129 IV . Young tableaux and combinatorial enumeration in parallel processing 136 References
139
Index
145 vii
Preface
In this volume we will present some applications of special functions in computer science . This largely consists of adaptations of articles that have appeared in the literature . Here they are presented in a format made accessible for the non-expert by providing some context . The material on group representations and Young tableaux is introductory in nature . However, the algebraic approach of Chapter 2 is original to the authors and has not appeared previously. Similarly, the material and approach based on Appell states, so formulated, is presented here for the first time . As in all volumes of this series, this one is suitable for self-study by researchers . It is as well appropriate as a text for a course or advanced seminar . The solutions are tackled with the help of various analytical techniques, such as generating functions, and probabilistic methods/insights appear regularly . An interesting feature is that, as has been the case in classical applications to physics, special functions arise - here in complexity analysis . And, as in physics, their appearance indicates an underlying Lie structure . Our primary audience is applied mathematicians and theoretical computer scientists . We are quite sure that pure mathematicians will find this volume interesting and useful as well . We expect this volume to have a utility between a reference and a monograph . We wish to make available in one volume a group of works and results scattered in the literature while providing some background to the mathematics involved which the reader will no doubt find appealing in its own right . The authors gratefully acknowledge AFCET, Universite de Nancy I, Southern Illinois University, INRIA, and NATO, for support while various parts of this work were carried out .
ix
INTRODUCTION
General remarks
In this volume, we present some applications of algebraic, analytic, and probabilistic methods to some problems in computer science. Special functions, notably orthogonal polynomials, and Bessel functions will appear. In Chapter J, we present the basic data structures we will be studying and introduce the ideas guiding the analysis. Chapter 2 presents applications of orthogonal polynomials and continued fractions to the analysis of dynamic data structures in the model introduced by J. Frangon and D.E. Knuth, called Knuth's model. The approach here is original with this exposition. The underlying algebraic/analytic structures play a prominent role. Chapter 3 presents some results involving Bessel functions and Lommel polynomials arising from a study of the behavior of the symbol table. Then, in another direction, theta functions come up as solutions of the heat equation on bounded domains, describing the limiting behavior of the evolution of stacks and the banker's algorithm. In Chapter 4 we present some basic material on representations of finite groups, including Fourier transform. Then Fourier transform is considered in more detail for abelian groups, particularly cyclic groups. Next we look at Krawtchouk polynomials, which arise in a variety of ways, e.g., in the study of random walks. The concluding Chapter 5 discusses representations of the symmetric group and connections with Young tableaux. Then, a variety of applications of Young tableaux are shown, including examples related to questions in parallel processing. Chapters 4 and 5 mainly include material from the literature that we find particularly interesting as well as some new approaches. It is hoped that the reader will find these chapters to be useful as background for further study as well as providing examples having effective illustrative and reference value. R e m a r k . The remainder of this introductory chapter provides some basic information that the reader may find useful, and is included so that the volume is fairly self-contained. Here one can find the notational conventions used in this volume; they are consistent with those of Volume 1. Many formulas and much information may be found in the Handbook by Abrajnowitz&Stegun [2], an overall very handy reference for the material of this chapter and for properties of the special functions used throughout this volume. Also, see Rainville[73].
2
INTRODUCTION
II.
Some notations
1. First, we recall the gamma and beta functions, given by the integrals r(x)= / Jo
t^-ifi-'dt (2.1)
r(x + y)
Jo
for R e x , Hey > 0. 2. In hypergeometric functions, we use the standard Pochammer notation:
(^)'^
r(a + 6) r(a)
where T denotes the gamma function. Thus, we have for binomial coeiEcients:
©=<-"'
A:!
3. For hypergeometric functions A ^ v ^ (ai)n(a2)n---(ap)n x"
a i , a ^ , . . . , dp
for integer p,q >0. For example, we have (fl)n(fe)«
2-ro
n
71 = 0
= 1 + a6a; + a ( a + 1 ) 6 ( 6 + l ) i V 2 + --4. For Stirling numbers, we use Sn,k and s^^k for Stirling numbers second kinds, respectively. They may be defined by the relations:
of the first and
(x)„ = Y^S„,kx'' k=0 'Tn
x" = ^ . „ , t ( - l ) " - * ( x ) , t=o See Knuth[53], p. 65fF. (where, however, a different notation is used). 5. For finite sets E, we denote the cardinality of E by \E\ or # E .
(2-2)
INTRODUCTION
III.
3
O r t h o g o n a l p o l y n o m i a l s and continued fractions
Orthogonal polynomials arise frequently as solutions to three-term recurrence relations. A standard reference on orthogonal polynomials is Szego[81]. For our work Chihara[10] is a useful reference as well. 3.1
ORTHOGONAL POLYNOMIALS
Given a bounded, nondecreasing function F{x) on R , we define a corresponding Stieltjes measure dF{x). The moments of dF are defined by
We will take F to be normalized so that ^o = 1- It is assumed that the moments exist for all n > 0. Integration with respect to dF, m,athem,atical expectation or expected value , is denoted by angle brackets ( • ) {f(X))= r
f(x)dF(x)
(3.1.1)
J — CO
in particular, we can write the moments as Hn = {X"). In the discrete case, we have a probability sequence (discrete distribution), {pk }k>o, satisfying p* > 0, ^ p ^ t = 1, with corresponding moments /^n =
Y^(ak)"pk k=0
where at denotes the value taken on with corresponding probability pk- In the (absolutely) continuous case, we have a nonnegative density function p{x), and fJ'n =
I x" J — oo
p{x)dx
We define an inner product (•, •} on the set of polynomials, which is the mathematical expectation of their product. On the basis { x" }, this is given in terms of the moments: {X",X"'} = ( X " . X ' " ) = /
X^-X"" dF{x) = fXn+m
J — oo
The corresponding sequence of orthogonal polynomials, { <j>„{x) }n>Oi say, may be determined by applying the Gram-Schmidt orthogonalization procedure to the basis { a;" }. The polynomials {
n{x) } satisfy M^)^m{x)dF{x) = n"'
^ , ^
(3.1.2)
4
INTRODUCTION
where 7„ are the squared norms. We normalize 70 to 1. With (f>„{x) = x" + • • •, monic polynomials, the { <j>„{x) } satisfy a three-term recurrence relation of the form: XTi=
0,nn-'rhn4>„-l
(3.1.3)
with ?io = 1, (^1 = X — ao. Orthogonality, comparing (a;i^„,(^„_i) = 6n7n-i with {4>n,x4in-\) = 7n, yields the relation: ln=Wh---hn
3.2
(3.1.4)
CONTINUED FRACTIONS
A general reference for continued fractions is Jones&Thron[47]. Two sequences of (real or complex) numbers { a„ }n>i> { ^n }n>o determine a continued fraction: 6o -I- ai hi -f- 02/62 + . . . + a„/fe„ -H . . . which is the limit as n —» oo of the
approximants:
—^ = 60 + 0.1 ^bi + 02/62 + • - • + a„/bn
P„ and Q„ are the n"" partial numerators and partiaJ denominators, respectively. They satisfy the recurrence relations: U„ = b„Un-l + anUn-2
(3.2.1)
with initial conditions P _ i = l , Po=6o,
g _ i = 0 , Qo = l
respectively. One may remark that this setup corresponds to the matrix relations:
5n Q
Pn
J
\an
b„J
\ai
bi J \1
bo J
INTRODUCTION
3.2.1 Connection
5
with orthogonal
polynomials
Consider the continued fraction 1 / l - aox - Wx^ j l -axx-
hix'^ j ... j l - a „ _ i z - 6„a;^ / . . .
(3.2.1.1)
From equation (3.2.1), the partial numerators and denominators satisfy "n+i = (1 - a.nx)un - &„a;^t<„_i for n > 1, where the numerator poJynomiais, P„{x), satisfy the initial conditions PQ = 0, Pi = 1, and the denominator polynomials, Qn{x), satisfy Qo = 1, Qi = 1 — aox. The reciprocal polynomials, j/'„(a;), defined as
M^)
=
^"Qn{l/x)
satisfy the three-term recurrence xtj^n = ipn+i + a„tl;„ + b„'ipn-.t
(3.2.1.2)
with initial conditions I/JO = 1, V"! = ^^ — tio- These give a sequence of orthogoned polynomials with squared norms 7n = &1&2 • • • 6„, n > 0. IV.
4.1
B e s s e l functions, L o m m e l p o l y n o m i a l s , t h e t a functions
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
The reader is referred to G.N. Watson's [84] treatise for an extensive presentation of the theory of Bessel functions. Here we indicate the features that are important for our applications. The Bessel function J„ is defined by :
..M = (./yE„'r'ritt) . . . . J r ( n + « + l)
<"•"
This function arises as the solution of the differential equation cPu
1 du
/.
v^
In the present study, a principal feature of Bessel functions is the fundamental recurrence: 2iyz-'j^{z)
= J^+i{z) + J.-i{z)
(4.1.3)
6
INTRODUCTION
Iterating this recurrence, Lommel polynomials come up as the coefficients expressing J^j^„(z) as a linear combination of Jv{z) and Jv-i(z)'.
This holds for nonnegative integer n. F o r n = 1, wehave/li,i,(z) = 2i^/2r, while i?o,i/(-2) = 1The Lommel polynomials satisfy the recurrence 2z-'^{n + v)R„=
R„+i + Rn-i
with the initial conditions i i _ i = 0, iio = 1- The i?„ may be given explicitly in the form [«/2] /
,x
, .
/„x
«..W^ g (";')(-)* ^ 0 )
n-2k
*:=0
They are used to solve the recurrence X ^ n = V'n+l + e ~ ' " V ' n + ^ n - l
(4.1.5)
which has as general solution i>„ - CiJ?„,_£^{-2e) - C2iZn-i,i-£2:(-2e) i.e., with initial values ^ _ i = C2,
= J-^^{-2e)R„,-exi-2s)
- J-i^,^{~2e)Rn-i,i-exi-2£)
which combines features of both eqs. (4.1.3), (4.1.4). 4.2
THETA FUNCTIONS
The four Jacobi theta functions
are given by
oo
^i{z,q)=
Y,
(-l)"9^"+^''^''sin(2n + l)2
? l = —OO OO
,?2(2,g)=
Yl
9^"+'^'''cos(2n + l)z
n = —OO
oo
^3(-^,?)=
2v
^"
cos2nz
n = —oo oo
M^^l)=
Y
(-l)"9"'cos2n^
(4.1.6)
INTRODUCTION
V.
7
S o m e analytic techniques
Here we indicate some basic analytic techniques that will be used. Included are: asymptotic behavior of coefficients of a power series, central limit approximation of sums involving binomial coefficients, and Lagrange inversion. 5.1
BASICS OF ASYMPTOTICS oo
Let f{z) — 2_^ Onz" be analytic in a neighborhood of 0 in the complex plane. We are n=0
interested in the behavior of |a„| as n ^ oo. The easiest observation is: oo
5.1.1 P r o p o s i t i o n .
Let f{z) = V^ ctn-^" be analytic in a disk of radius R > 1. Then n=0
the coefficients an decrease exponentially
fast.
Proof: The terms of the series 2 ^ Q |c(„|(iJ — e)" converge to zero for any e > 0. Choose e so that R~ e > 1. • Because of this, one looks for the singularities of functions when it is known that the coefficients a„ grow as n —» oo. For our purposes, the following suffices. 5.1.2 L e m m a . Let f{z) = S ^ o * * " ^ " ^'^ meromorphic, analytic in a neighborhood of 0 in the complex plane. Let ^ denote the pole of f nearest to the origin. If ( is a poJe of order p, then we have the estimate ,
,
A
nP-i
r(p) c"+p where A = lim f(z){z
Proof:
— C)*".
Write g{z) = f{z) • {z - CY- So A = g{0 and
where the second term has at most a pole of order < p. Now, for \z\ < |C|, the binomial theorem implies
At a pole of larger modulus, the same estimate gives a rate of growth exponentially slower, • whereas terms of lower order poles grow at correspondingly lower powers of n. See Flajolet-Vitter[28] for more along these lines.
8 5.2
INTRODUCTION CENTRAL LIMIT APPROXIMATION
The central limit approximation is a detailed application of the theorem of DeMoivreLaplace that one can approximate sums involving binomial coefficients, interpreted as taking expectations with respect to binomial distributions, by Gaussian integrals. Here is a useful version. 5.2.1 L e m m a .
Then, as N —* oo,
Let f be a polynomial.
Another formulation of the same statement is:
N \ /' N k + N/2) ^ \N/2 where k = avN.
In particular, we have
5.2.2 L e m m a .
For p > 0, as iV —> 00,
2-2"'
y/Nda
2 / ^ 1 / 1
.,4-',,^ |*|
{k + N/2j ^ ' •"
\NI2) \ ' /
»
^
^V
This follows from the above remarks and the Gaussian integral J.OO
oo
/
a'^'e-^'^" da = 2 -00
a^^g-^"' da Jo
= 7rTT7^ /
t^-'^'e-'dt
2P+1/2 y^,
= ^ r ( p + i/2) via the substitution t = 2a^. The result follows, using r ( | ) = y/n.
•
See Canfield[9] for more information on this topic. 5.3
LAGRANGE INVERSION
Given a function, / , analytic in a neighborhood of XQ, that is locally one-to-one, write y = f{x). Lagrange inversion provides a way of finding expansions of functions of x in terms of the variable y. In particular, the relation x = f~^{y) is the inverse of / . The idea is that we want the expansion in terms of y of any function g{x) analytic in a neighborhood of xo.
INTRODUCTION
5.3.1 P r o p o s i t i o n . Let y = f{x) be anaJyfic near XQ, yo = f{xo), with / ' ( x o ) 7^ 0. For g analytic near xo, the Lagrange inversion formula gives the expansion k-\
^
k=l
See Sansone-Gerretsen[79] for details. integration by parts. VI.
I,
s
'
I
Xo
X
fix)
- yo
The proof uses Cauchy's integral formula and
P o l y n o m i a l s : reference information
Here we include some basic reference material on the classes of orthogonal polynomials that come up, notably in the study of dynamic d a t a structures. In general, references are Szego[81] and Chihara[10]. First we note 1. The inner product is denoted by angle brackets (•,•), thus f(x)gix)dFix) -00
In particular, for an orthogonal sequence {„}, we have {n,4>m) = j
'i>nix)(j>mix)dF{x)
=Jn6„m
J —00
where 6nm is Kronecker's delta equal to 1 if n = m, 0 otherwise, and 7„ are the squared norms. 2. The moment generating function denotes a generating function for the sequence of moments {^n }• Note that for symmetric distributions, where the odd moments vanish, we often write the generating function for { n^n }• 6.1
POLYNOMIALS CORRESPONDING TO DISCRETE DISTRIBUTIONS
Here we list basic information on polynomials orthogonal with respect to the binomial and Poisson distributions, respectively. • Binomial distribution — Krawtchouk polynomials { Kn{x) } Recurrence formula X Kn - A'„+i + n{N - n + l)A'„_i with A'o = 1, K\ = X.
10
INTRODUCTION
Measure of orthogonality and squared norms
{Kn, Km) = 2-^^ ^
jn =
( , )K,,{2k
- N)Kmi2k
- N) = y„ 6„^
n\{-in-N)„
Moment generating function V —r/^n = (cosh 5 ) ^ Generating function for the polynomials N
T ^ K„(x) = (1 + t,)(^+^)/^(l - vY^-^y' n=0
Formula for the Krawtchouk polynomials
lt=0
• Poisson distribution — Poisson-Charlier polynomials { P„{x,t) } Recurrence formula xPn=
Pn+l
+ (< + n)Pn
+
tuPn-l
with Po = 1, Pi = a ; - t . Measure of orthogonality and squared norms {Pn, P,n)
=e-'
^ 4=0
-
Pnik)Pm{k)
= 7 „ Sr,
•
7n = n\r Moments /in =
Y^Sn^kt'' k
Moment generating function V n=0
—Ain = e x p ( i ( e ' - 1 ) ) n
INTRODUCTION
11
Generating function for the polynomials
n=0
Formula for the Poisson-Charlier polynomials
P„(x,0 = (-ir^(")<"-*(-a.)* jfc=o ^
^
R e m a r k . The moments for the Poisson distribution are readily calculated by considering /•oo
/
oo
y^ dF[x) = e-*
V k=0
Differentiating n times with respect to j / , then setting y — 1, yields / Jo
xix-l)---{x-n
+ l)dF{x)=
( - l ) " ( - x ) „ d F ( T ) = <" Jo
Then the Stirling numbers of the second kind, eq. (2.2), convert these to the usual moments given above. 6.2
POLYNOMIALS CORRESPONDING TO CONTINUOUS DISTRIBUTIONS
Here we have Tchebychev and Hermite polynomials. Recall that the Tchebychev polynomials are determined according to T„(cos $) = cos nO,
Un{cos 9) = sin(n + \)9/ sin 6
The Hermite polynomials are orthogonal with respect to the Gaussian distribution. • Tchebychev polynomials of the first kind { T„(a;) } Recurrence formula 2x Tn = T„-^.l + Tn-i with To = 1, Ti = X. Measure of orthogonality and squared norms
1
f^
dx
TT J-1
7n = 2 '
V l
^^'^
-
X^
12
INTRODUCTION
Moments
Moment generating function
yr^
n=0
Generating function for the polynomials 1 — XV
^'"^'M-T^^^. 71 = 0
Formula for the Tchebychev polynomials of the first kind [n/21
• Tchebychev polynomials of the second kind { U„{x) } Recurrence formula 2xUn with Uo =l,Ui
=U„+i+Un-l
= 2x.
Measure of orthogonality and squared norms 2 {Un,Um)
yi
—— /
7n = 1 ,
Uriix)Um{x)\/l
n>Q
Moments
Moment generating function 2^ S
/^2n - 2
Generating function for the polynomials
- x'^ dx =
'y„S„„
INTRODUCTION
13
Formula for the Tchebychev polynomials of the second kind [n/2] k/n^\n-2k *:=0
^
^
• Gaussian distribution — Hermite polynomials { Hn{x,i) Recurrence formula xH„ = Hn+i +
inHn-i
with Ho = 1, ffi = X. Measure of orthogonality and squared norms
-oo
7n = n\t" Moments n,l^
^2n = ( 2 0 " ( | ) n = Moment generating function
n=0
Generating function for the polynomials
. n! Formula for the Hermite polynomials [n/2] ^ ' / r) \ k=0 ^
^
( 2 » ) ! .n
^ ^ <
}
Chapter 1
BASIC DATA S T R U C T U R E S
In this Chapter are introduced the basic data structures that will be the principal subject of the first two chapters (and part of Chapter 3). First, we introduce basic terminology and constructions. In Chapter 2, detailed computations and results will be presented. The structures discussed here are representations of files or lists of data stored in a computer. The idea is to organize the data for ready access and effective use of the information. I.
Basic d a t a s t r u c t u r e s
We start with a universe of elements, items. It is assumed that one can equip the universe with a total order. Once given an order structure, the items are referred to as keys 1.1 Definition.
A list
is a sequence of items.
When the items can be ordered, i.e., when the list consists of keys, then one has a sorted or unsorted list accordingly. Particular data types are determined according to operations used to construct corresponding lists. A sequence of operations performed on a given structure is represented as a word in a language with alphabet corresponding to the operations allowed to be performed on the data structure. Here we consider structures which permit insertions denoted by J, deletions , denoted D, and record searches (examining items for information) called queries denoted Q"*" or Q~ according as the result of the search was positive (successful) or negative (unsuccessful). This gives a basic alphabet {I,D,Q+,Q-}. R e m a r k . Since the words can be of arbitrary length, one could consider the set of infinite sequences of letters, restricting to the subclass of tame sequences, having all but finitely many components equal to e the 'empty symbol', denoting a blank, which thus would be adjoined (formally) as part of the alphabet. 1.1
LINEAR LISTS AND DICTIONARIES
A basic structure of interest is the linear list. This is a list which is constructed using only insertions and deletions. 1.1.1 Definition.
A linear list
is a list that admits the operations / and D only.
For purposes of illustration, we denote the items by lowercase letters { a, 6, c , . . . } and the corresponding operations as a word using the alphabet { I,D}. For example, starting from abc, inserting d, e, / , then deleting a and / might give successively: dabc, dabec, dabfec, dbfec, dbec, corresponding to the word IIIDD. It is clear that many different lists may correspond to a given sequence of operations. A main feature of the analysis is how to deal with precisely this fact in determining the complexity of a particular structure. Probably the simplest and most well-known example of a lineax list is the stack.
BASIC DATA STRUCTURES
15
1.1.2 Deflnition. A stack is a linear list for which the insertions and deletions are all made at only one end of the list. For a stack, a given word corresponds to only one possible set of consequences. In the above example, after three insertions, the deletions would necessarily indicate removal of / and then e from the list. 1.1.1
Queues
Another important class of structures are queues. 1.1.1.1 Definition. A queue is a linear list for which insertions are made at one end of the list and deletions are made at the other end. As with a stack, a series of operations corresponds to only one possible set of consequences. An important modification is the priority queue. We assume that the items are keys, i.e., come from a totally ordered universe. In the priority queue, only the highest priority item can be deleted, with priority according to the ordering of the keys. 1.1.1.2 Definition. A priority queue is a linear list of keys for which only the minimal key may be deleted. Insertions are unrestricted.
1.1.2
Dictionaries
Dictionaries are modelled on the everyday notion of dictionary, i.e., queries, corresponding to looking up information, arise. Thus, all four operations I, D, Q"*", Q~ are admitted. The elements are keys. 1.1.2.1 Definition. Q+, Q--
A dictionary
is a list of keys admitting the four o p e r a t i o n s / , Z),
A queue/dictionary structure that gives rise to an especially interesting mathematical analysis is the symbol table. Here positive queries are admitted, but only the last key inserted may be deleted. I.e., 1.1.2.2 Definition. A symbol table is a list of keys admitting the three operations / , D, Q"*". Deletions operate only on the last (most recent) key inserted.
At this point it is not clear which structure is the most effective in a given situation. The question is how to determine what practical implementations are most efficient. In the next section we consider some basic data organizations — how the structures are implemented in the machine. The mathematical analysis, average case analysis, gives one an idea of the relative efficiency of the various data organizations.
16 1.2
CHAPTER 1 IMPLEMENTATIONS
We will be concerned primarily with linear lists, priority queues, dictionaries, and symbol tables. Each of these data types may be implemented as sorted lists, for keys, as well as unsorted lists. Additionally there are various implementations available for these structures. Below, we illustrate the use of pagodas, a structure developed by Fran^on, Viennot& Vuillemin [35]. R e m a r k . Sorting algorithms are a good example of the differences that can arise between worst-case and average-case behavior. Consider the most efficient sorting algorithm, 'quicksort', developed by C.A.R. Hoare. Analysis shows that this algorithm sorts a sequence of n elements in an average time of the form O ( n l o g n ) , i.e., bounded by a constant times n log 77. Comparison among all sorting algorithms yields the conclusion that quicksort is the most efKcient on the average. However, the worst case, when the data is almost in order at the start, is quite poor, taking on the order of n^ steps. Fortunately these worst cases are rare. See Knuth[53], v. 3, p. 114ff. For priority queues, in addition to sorted and unsorted lists, there are implementations as binary tournaments and pagodas. The idea of these structures is to take advantage of the priorities among the items. Here we identify a priority queue of size n with a permutation of the integers from 1 to n, the keys. To these correspond labelled, rooted trees. The labels are the keys 1 , . . . , n. First we recall the definition of a binary tree . A binary tree T is given recursively as a triple T = {l{T),v{T),r{T)) where l(T) and r{T) are binary trees, the left and right subtrees respectively of T, and v(T) is the label (value) of the root of T. The empty tree is included in the definition of a binary tree. Similarly, a binary tournament is defined recursively by the condition that the left and right subtrees of the root are binary tournaments satisfying v{T) < min{ v{(): ( a node of 1{T) or r{T) }. In other words, for any node (, its sons (' satisfy v{() < v{^'), priority increasing with the depth of the node. R e m a r k . Various types of tournaments have been designed depending on the constraints imposed. Heaps, AVL trees, 3-2 trees, binomial queues are among the most well known. See Aho, Hopcroft&; Ullman[3], Knuth[53], Mehlhorn&Tsakalidis[68] for full discussions. For example, consider the queue (permutation) 573961428. Successively taking out the minimal values, partitioning accordingly, and otherwise keeping order, leads to the following binary tournament: 1 /
\
3
2
/
\
^
5
6 \
^ 7
9
4
\ 8
BASIC DATA STRUCTURES
17
or, adjoining O's so that every node has left and right subtrees, this can be represented by the set of triples a = (c,l,6)
d = (0,4,0)
g=
(0,7,0)
h={d,2,h)
e = (0,5,5)
/i = (0,8,0)
c = (e,3,/)
/=(»,6,0)
i = (0,9,0)
For example, 6 = (d, 2, h) indicates that the node 2 has attached left subtree d ajid right subtree h. 1.2.1 Pagodas Now we are ready for the construction of pagodas. Start with a binary tournament T, given as a set of triples (as above). The pagoda representation is a set of triples determined as follows. For each subtree P, determine the left inverse subtree l''^{P) as the unique subtree, Q, say, such that 1{Q) = P in the triple corresponding to Q. For example, in the tournament of the previous section, we have /~^(e) = c. Similarly, determine the right inverse subtree of P. Next, define recursively the leftmost descendant of a subtree P as follows i*( p> _ / -P' if KP) is empty I ^*{KP)), otherwise r*{P) is defined correspondingly. (Here, empty subtrees correspond to the O's in the triples used in the representation given above for tournaments.) Now we have 1.2.1.1 Definition. Let T be a binary tournament. A •pagoda is a representation of T as a set of triples (A(P), w(P), p{P)), one for each nonempty subtree P of T, such that \(p\^i^~\P)^ ^ ^ \l\P),
if r ^ P ) exists otherwise
p(P)=(''~'(-P)' r*[P),
i f ' - " H ^ ) exists otherwise
and
Briefly, for each subtree locate its (left or right) ancestor if there is one; if none, then give its furthest descendant. For example, the pagoda representation for the tournament considered above is given by: a = ( e , l,/i)
d=(6,4,d)
g = {g,7,e}
b = (d, 2,a)
e = (c, b,g)
h = (h, 8, ft)
c=(a,3,/)
f = {i,6,c)
t =
{f,9,i)
Frangon, Viennot&Vuillemin[35] proved that, for implementing priority queues, the pagoda has the best average complexity (see §1.5 of Chapter 2).
18
II.
CHAPTER 1
Dynamic data structures
The motivation for studying dynamic data structures is to compare the efficiency of the various data structures as implemented. How is one to analyze the behavior of these systems? Two basic ideas are time costs and space costs, involving respectively the complexity due to the number of operations involved and the complexity due to storage — memory — requirements. Here we study time costs. 2.1
HISTORIES
We present a useful formalism. Let O denote the alphabet of operations, in our case we have O C {I,D,Q'^,Q~ }. Thus, a word means a sequence of elements belonging to O. We denote the set of words over the alphabet O hy O*. We want to count the number of operations involved when the operations are performed on files under constraints imposed according to the various data structures. First, we make a general definition, corresponding to the necessary condition that a file cannot be of negative size. 2.1.1 Definition. appears in w.
For w £ O* and O £ O, denote by \w\o the number of times O
2.1.2 Definition.
For w e C*, the height
of w,/i(w), is defined by
h{'w) = \w\i -
\W\D
Note that the difference between the number of insertions and deletions is just the size of the file. Now we have 2.1.3 Definition. A schema is a word w = O1O2 • • • On € O* such that the sequence of subwords Wi = O1O2 • • -Oi satisfies h(wi) > 0, for all i, 1 < i < n. When counting file operations involving keys, we consider only their relative order. The method of counting the number of ways an operation can be applied to a given file in a given situation is determined by a sequence of nonnegative integers, ranks, corresponding to (the relative ordering of) keys. 2.1.4 Definition. Let ri,...,r„ be a sequence of nonnegative integers, ranks . A history is a sequence of ordered pairs { ( O i , r i ) , . . . ,{0„,r„) } such that O1O2 • • On is a schema and r j , . . . , r „ are admissible ranks (for a given model). The ranks allowed determine the particular models. Below, we discuss two models that have been studied: the maricovian model and Knuth's model. So a model is determined by the alphabet O and, for each data structure, the specification of allowed histories. (In Chapter 2, detailed results for Knuth's model will be presented.) Here we continue with a sketch of the general approach, assuming a model has been given.
BASIC DATA STRUCTURES
19
2.1.5 Definition. For an operation O, the number of possibilities number of ranks r for which O is defined on a file of size k.
N p o s ( 0 , fc) is the
If an operation is not allowed at a given time, we set Npos (0,k) ~ 0. This determines the count of the number of operations of a given type that are permitted to be performed on a file. 2.2
TIME COSTS
First some notation. 2.2.1 Definition.
For a given class of histories:
1) '^k,t,n denotes the set of histories of length n with initial height k and final height /. Denote the cardinality |?it,;,„| by Hk,i,n2) Hn denotes the set of histories starting and ending with an empty file, i.e., Tin = 'Ho,Q,n- Denote the cardinality |7f„| by H„. Suppose we are given a cost functional on the set of histories: cost (h), some measure of the complexity of the histories. Then we have 2.2.2 Definition.
The integrated cost 1 iiTn = —
ov6r 7^n is ^
cost (h)
In order to analyze time costs, it is convenient that the data structure satisfy a condition of stationarity. Many implementations commonly used in computer science have the required property. Here is one way of stating what is needed. Denote by Ek the set of all states of a structure of size k. This depends on the particular implementation. For example, for a sorted list, there is only one state for each fc, while there axe A;! possible states of size k for an unsorted list. As another example, there are k\ pagodas of size k. One defines 2.2.3 Definition. The standard probability distribution on Ek is determined by giving equal weight to all states of size k corresponding to the schema /*. We denote by Ps(e) the corresponding probability of a state e £ Ek-
2.2.4 Definition. A data structure is stationary if for all k, probabilities induced on Ek by histories corresponding to schema /*"^^D, l''Q~, and l''Q'^ (respectively) coincide with the standard probability distribution p,. The structures we will consider are known to be stationary. This property permits the calculation of costs in a particularly effective manner. Namely, we can use the notion of individual costs.
20
CHAPTER 1
2.2.5 Definition.
The individual cost COk=
^
of operation O on a file of size k is given by ps{e) cost
{0,e)
e€Ek
where cost (O, e) is the cost of performing operation O on the state e. Here costs are determined by the average number of comparisons of keys required for the operation. The following table gives the average costs of operations for various implementations, (In this table we use the notation /li = 1 + | + • • • + p )
Data type Linear list
Priority queue
Dictionary
Implementation Sorted list Unsorted list Sorted list Unsorted list Binary tournament Pagodas Sorted list Unsorted list
CI;
CD,
0
t+1 2 fc+1 2
CQt
CQ,
k+2 2
k+1 2
k+2 2
0
b±l
k
k+2 2
0
2^1
0
k-1
2(;.. + i ^ )
2
Symbol table
Sorted list Unsorted list
k - 1
0
0
k+1 2 k+1 2
From this table we can see that the relative efficiencies are not evident. For example, look at dictionaries. The table shows that, implemented as a list, the average cost of an insertion in a dictionary of size k is (k + 2)/2 if the hst is sorted, 0 if unsorted. The average cost for deletions and positive queries is independent of whether the list is sorted, while the average cost of a negative query is (A;-|-2)/2 for sorted lists, k for unsorted. Comparing these costs, it is unclear how to determine which list structure is more efficient. This is where the idea (of Frajigon, Knuth, et al.) to consider the dynamic behavior of these structures comes from. To compute the average (integrated) cost we use the individual costs COk- First we have 2.2.6 Definition. The level crossing number NOk,n is the number of operations of type O performed on a file of size k in the course of the histories W„.
BASIC DATA STRUCTURES
21
We can thus write n-l
NOt,r. = Npos {Ok) • Y, -^o.t-' Ho,k',„-,-i
(2.2.1)
1=0
where k,
for 0 = Q+ OT Q-
it + 1, /b - 1,
for O = J for O = JO
! From the definitions, we see that 2.2.7 P r o p o s i t i o n . For stationary structures, the integrated
cost is given by
^'" = TT E E '^^>= • ^^*." 0€0
2.3
k
POSSIBILITY FUNCTIONS: MARKOVIAN MODEL, KNUTH'S MODEL
The models presented here differ principally in the way one counts insertions (and negative queries). In [54], Knuth discusses two types of insertions in a data structure. Take the interval [0,1] as the set of keys. Consider inserting the next key into a file of k keys: /o means insertion of a random number by order in the sense that the new number is equally likely to fall in any of the k + 1 intervals determined by the keys in the file / means the insertion of a random number uniformly distributed on the interval [0,1], independent of previously inserted numbers. G.D. Knott[52] showed that Ig differs from / . Considering insertions of type IQ gives the marJcovian model (studied in Fran5on[29]). 2.S.1 Markovian
model
For this model the specification of histories is as follows (recall that h{w) is the height of the word w, i.e., it is the size of the corresponding file): 1. For linear lists, the alphabet of operations O = {I,D] ing: 0
with admissible ranks satisfyOj = D Oj = I
2. For priority queues, C = { / , Z? } with admissible ranks satisfying: r, = 0 0
0,=D Oj=I
22
CHAPTER 1
3. For dictionaries, O = {I,D, Q"*", Q~ ] with admissible ranks satisfying: 0
Oj=DoiQ+
0
OJ=IOTQ-
4. For symbol tables, O = { I,D,Q^
} with admissible ranks satisfying:
Tj = 0
Oj=D
0
Oj = Q+
0
Oj=I
Thus, the possibility functions for this model are: Datatype Linear list Priority queue Dictionary Symbol table 2.S.2 Knuth's
Npos(7,A;)
Npos(Z?,fc)
k+l k+l k+l k+l
k
Npos(Q+,A:)
'Npos{Q-,k)
1
k 1
k+ 1
k k
model
Considering insertions of type / gives Knuth's model. Here is an example of how this works for a linear list. Consider the sequence of operations IIIDI, starting from an empty file. Let X < y < z he the first three keys inserted. We have x, y and z equally likely to be deleted. Let w be the fourth key inserted. Whichever key has been deleted, considering all four keys, the four possibilities: w<x
with admissible ranks satisfying:
0
Oj = D
0
o, = i
BASIC DATA STRUCTURES
23
2. For priority queues, O = {I,D}
with admissible ranks satisfying:
rj = 0 0
Oj = D
<\Oi...Oj-i\i
Oj = I
Q"*", Q~ } with admissible ranks satisfying:
0
Oj=D
0
Oj = / or Q -
4. For symbol tables, O = {I,D,
Q"*" } with admissible ranks satisfying:
rj =0 0
ov Q+
Oj = D Oj = Q+ 0,=I
Nposfi"" 7 or Q - )
Npos(Z?,/fc)
i i i
k 1 k i
l
Npos(Q+,A:)
k k
We will discuss results for this model in Chapters 2 and 3. R e m a r k . As basic references for the subject; Gormen, Leiserson&Rivest[12], Gonnet&Baeza-Yates[39], Graham, Knuth&Patashnik[40], Greene&Knuth[41], Hofri[45], Kemp[50], Mehlhorn[67], Mehlhorn&;Tsakahdis[68], and Sedgewick[80]. A basic text is Aho, Hopcroft&Ullman[3]. As a general reference, see Fair, Mohr&Schott[71]. For an overview, see the handbook edited by van Leeuwen[56]. Of course, Knuth[53] is a general reference.
Chapter 2 DATA S T R U C T U R E S & O R T H O G O N A L POLYNOMIALS
In this Chapter, we first present the analysis involved in finding the behavior of data structures in Knuth's model. Then we indicate some features of a finite universe model, which has perhaps a more realistic flavor than the models first discussed. Then we consider mutual exclusion. This models a system of parallel processes accessed, at random, by various users. Each process can be accessed by only one user at a time, hence the description. A probabilistic model again leads to recurrences of the type studied in this Chapter. We conclude with some remarks on a general approach to duality, relating the space-time recurrence with the dual action of the operator of multiplication by X. I.
Dynamic data structures: Knuth's model
Here we will analyze the behavior of the data structures of Chapter 1, §2.3.2. We present the approach based on path counting, used as well in the study of random walks. Using the notion of dual recurrences this leads to orthogonal polynomials and an associated operator calculus. We then employ the operator calculus in our analysis of the data structures. We will look in detail at the behavior of histories and corresponding integrated costs for linear lists, priority queues, and dictionaries, following the indications of Ch. 1, §2.2. First, we enumerate the histories. Then we obtain expressions for the integrated costs using level crossing numbers and the individual costs. This puts us in a position to compare the relative efficiency of different implementations. 1.1
ENUMERATION O F HISTORIES AND ORTHOGONAL POLYNOMIALS
Before looking at Knuth's model, we introduce the basic technique in the context of the markovian model, where the approach is directly applicable. Then we will return to Knuth's model. Here we use the approach used in the study of random walks, where transitions from state (position) to state are governed by certain probabilities. For counting paths, the idea is the same, but we use the number of possible transitions, rather than probabilities. Consider the markovian model of Ch. 1, §2.3.1. Let c„t denote the number of paths, i.e., histories, starting from height 0 that are at level k at time n. This is, in fact, the same as Ho,k,n- We think of the height, file size, as the location of a particle (random walker) on the line. It jumps right or left or sits at each time step according to whether the operation 7, D, or a query is applied to the file. Denote ifc = N p o s ( / , A:),
qii =Npos{Q,k),
dk —
Npos{D,k)
Then we have, considering the one-step transition from time n to time n -(- 1,
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
25
1.1.1 P r o p o s i t i o n . Let Cnk denote the niimber of ways starting from level 0 to reach level k in n steps. The c„jt satisfy the recurrence Cn+l k = U - l Cnk-l
+ qk Cnk + ^ * + l C„ k+\
with Coo = 1, Cot = 0, A: > 0. We can identify this recurrence with the transition
(
matrix
A
90
«o
0
rfi
91
«1
0
•••
0
d2
?2
22
• • •
0
0
dz
93
'••
0
0
0
d^
0
0
0
0
0
• • •
(1.1.1)
V where the kl entry indicates the number of ways of going from level k to level / in one step, fc, ? > 0. The n'*' power of this matrix thus gives the number of ways of going from one level to another in n steps. Now the idea is to look at the duaJ recurrence. Think of the matrix in (1.1-1) as an operator JT on a space with basis { <^o, 0 i , . . . , 4>k-, • • • }. Consider the spectral problem where X acts as multiplication by the variable x. Writing $ = (<^0i i^ij • • •) we want
This turns into the three-term recurrence relation for the sequence {i^t }: ^4>k= ik'i>k-^\ + (Ik^k + dk<^k-\
(1.1.2)
with initial conditions 4>-\ = 0, <j>o = 1. Thus, the {<^jt } are identified as orthogonal polynomials in the variable x. As in eq. (3.1.1) of the Introduction, we denote integration with respect to the corresponding distribution, expected value, by ( • ). The operator X is the one-step transition operator and may be thought of as a position operator (as in quantum mechanics). Another way to see that the recurrence relation (1.1.2) is dual to that of Prop. 1.1.1 is given by: 1.1.2 P r o p o s i t i o n .
The action of X on the basis { I/>A } satisfies X"0 = /^Cnkk fc=0
26
CHAPTER 2 Proof:
Inductively, assume the relation, which holds for n = 0. Then X""'"Vo = ^c„i(it(/>jfc+i + qkk + dk4>k-\) k = 2 _ , ( * * - l '^nk-l k =
by Prop. 1.1.1.
+ qkCnk + dk+\
Cnk+l)^k
}_CnJr\kk k
•
This shows X " as the n-step transition operator. Shifting all indices by k 1.1.3 Corollary. Denoting in n steps, we have
hy Cni,k the number of ways of going from level k to level I n
X''^k
= Y,Cnl,k(f>l 1=0
with coi,k = 1, ifi = k,0
otherwise.
We can introduce the monic polynomials J/J^ = ;t ioh • • • ik-irelation xil^k = i>k+i + Qki^k +
They satisfy the recurrence
ik-idki>k-i
Equation (3.2.1.2) of the Introduction indicates the link with the continued fraction approach of section §1.6. As follows by equations (3.1.3), (3.1.4) of the Intro., the squared norm of ^k is Ik — toil • • • ik-i
didi
•• -dk
As in §3.1 of the Intro., eq. (3.1.2), for the inner product we have 1.1.4 P r o p o s i t i o n .
The {(fek)
satisfy = k
«'•«-{?• j ^k with the squared
norms 2
ek = Ukr =
dxu2 * • • dk iQii
••
-ik-i
Thus, the markovian model indeed corresponds to the notion of Markov chain or random walk in this case. Let us rewrite the above corollary in terms of histories. We have the result (Proposition 7 of Flajolet, Fran^on &Vuillemin [26])
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
1.1.5 T h e o r e m .
In
The number of histories Hk,i,n is given by the expected
27
value
particular, Ho,k,n = {X"^k}/£k,
Hk,o,n = {X"<j>k}
with Ho,0,n = {X"') = fin the
moments.
And we have the observation, from the point of view of operator calculus, via the first statement of the Theorem, 1.1.6 Corollary.
The exponential
V
generating
^Hk,i,n
functions
= (e'^ k(t>i)/ei
are the matrix elements of the operator e'^
in the basis {(i>k}-
R e m a r k . This feature is a connection with Volume 1 [21] of this series, Chapter 7 in particular, that will be useful in our analysis. It turns out that, appropriately modified, these techniques can be applied to the study of Knuth's model as well. 1.1.1 Enumeration
of histories in Knuth's
model
Here we introduce c^j., the number of paths starting from 0 that are of height k after n steps, with s the number of insertions and negative queries combined. Prop. 1.1.1 becomes 1.1.1.1 P r o p o s i t i o n .
The c^f. satisfy the
recurrence
< + i k = ^ < " * - ! + s c^-1 + qk c;^ + dk+i c'„ ;t+i with CQO = 1, CQJ. = 0, A:,3 > 0. The idea here is to introduce the generating function
Cnk{t) = Xll"*^"* 8= 0
Thus, the s is summed out, leading to a situation similar to the maxkovian model. We see immediately that
28
CHAPTER 2
1.1.1.2 P r o p o s i t i o n .
The C„kit) satisfy the
recurrence
Cn+l k = tCn k-1 +{t + Qk) C„k + dk+\ Cn k+\ with Coo(t) = 1, Co kit) = 0, ^ > 0. The t indicates t in the case when there axe negative queries, otherwise it is omitted. Here we introduce the operator Xt acting on a space with basis { (j>k{x,t) } such that n
k=0
as in Prop. 1.1.2. Dual to Prop. 1.1.1.2 is 1.1.1.3 P r o p o s i t i o n . The polynomials { 4>kix,t) } satisfy the
recurrence
X(j>k = t(f>k+i + (t + qk)4>k + dk4>k-l with <^_i = 0, (/"o = 1. The squared norms are given by ekit):=Ukf
=t-''did2---dk
The corresponding monic polynomials {«/"*:} are given by tpk = <* k satisfying the recurrence -\-tdk^k-\ (1.1.1.1) xi/^k = V't+i +(i+qk)tpk The n-step transitions satisfy n
As in Theorem 1.1.5 we have 1.1.1.4 P r o p o s i t i o n . Let H^ ; „ denote the number of histories from level k to level I in n steps with s insertions and negative queries (combinedj. The generating function °° t" 3=0
satisfies the relations of Theorem 1.1.5 with t-dependent
X and { 4>k }-
To get back to the number of histories from HI, „(<) we observe the following 1.1.1.5 P r o p o s i t i o n .
The number of histories Hk,!,n is given by the integral Hk,l,n=
t-* Hli^^{t)dt Jo In particular, for histories starting and ending with an empty file Hn= with H„it) = Hofi,-a{t).
I Jo
e-'Hn{t)dt
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
29
Proof: In the above Proposition, integrate both sides with respect to e~* from 0 to oo using the relation /
t'e-^ dt = si
Jo The result on the right-hand side is
oo
3=0
which takes into account all possible insertions and deletions.
1.2
•
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS IN KNUTH'S MODEL
Now we apply the above results to some particular data structures. 1.2.1 Priority
queues
As noted above in Ch.l, §2.3.2, there are no queries, and dk = I, for all k > 1. Prop. 1.1.1.3 gives, where we rec?i,ll that the middle term, involving (pk, corresponds to queries: 1.2.1.1 P r o p o s i t i o n . 1. The recurrence
For priority queues we have:
relations X<j>k = i(f>k+l + k-\
with the squared norms ek{t) = <~*. 2. We have the
identification 4>k{x,t)=t-''I^Uk{xl2yft)
Tchebychev polynomials
of the second kind, and the moments
/^2n(0 =
<"
f2n
n + 1 \n Proof: The recurrence formula follows directly from Prop. 1.1.1.3, modified according to the absence of queries. Then # 2 follows from the recurrence formula for the Un given in Intro., §6.2. The density function for the orthogonality relations for the U„ is - \/l — x'^ on [—1,1]. Changing variables x —> x/(2y/i) gives the density
TTV^
on [—2\rt,2\/t].
V
it
The moments scale inversely as x, yielding iJ.2n{t) as indicated.
For the histories;
30
CHAPTER 2
1.2.1.2 P r o p o s i t i o n . is given by
The generating function for the number of histories,
Ho,k,nii),
rk)/2
tor 0 < k < n, k = n mod 2.
Proof:
The fl'o,jc,n(<) are the same as Cnk{t), satisfying Cn+lk = tC„k-l
+ Cnk+l
with Coo{t) = 1, Cok{t) = 0, k > 0. (Notice that this is very close to Pascal's triangle.) Inserting the formula in the statement of the Proposition, one sees that it must be verified, after cancelling common factors of i("+*:+i)/2^
n+2
\k
+ l
^(n + 1 - fc)/2y n + 2
n+1
(
\
^
V(" + 1 - ^ ) / 2 / « + 1
(
"+ 1
\ { n - \ - k)l2J
Writing out in terms of factorials, the result readily follows.
\k^-2 n +1
•
1.2.2 Linear lists Here, a.s in the previous section, there are no queries, while dk = k, for A: > 1. Thus, 1.2.2.1 P r o p o s i t i o n .
For linear hsts we have:
1. The recurrence reiations x4>k =t 4>k+i + k4'k-i with the squared norms £fc(<) = <^* k\ . 2. We have the
identification <j>kix,t)=t-'=Hk{x,t)
Hermite polynomials,
and the moments
fi2n{t) — (2i)" (|)TI .
Proof: The recurrence formula follows from Prop. 1.1.1.3, modified according to the absence of queries, and # 2 follows from the recurrence formula and facts given in Intro.,
§6.2.
•
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS 1.2.3
31
Dictionaries
Proceeding as in the above two sections, we find 1.2.3.1 P r o p o s i t i o n . 1. The recurrence
For dictionaries we have:
relations (j>k+\ + (< + k)(jik + kcpk-\
x(j>k-t
•with the squared norms et(<) = t~'' k\ . 2. We have the identification Mx,t) Poisson-Charher
1.3
polynomials,
=
and the moments
t~''Pk{x,t) Hn{t) = Hn = 2 . •*">* * •
OPERATOR CALCULUS APPROACH
Look at the dual recurrence showing the operator X acting on the basis vector (fije X4ik = «*<j^jfc+i + k + dk4>k-\
Each term of this recurrence corresponds to an action on the file: insertion, query, deletion. We define corresponding operators /Z, the raising operator, N, or neutral operator, and L, lowering operator, named according to their effect on the index k: R<j)k = ikk+i N4>k = qk(l>k L<^k = dk4>k-i
A useful operator V is defined by V(l>k =
{klik-i)4>k-\
so that RVH
= kcj>k
is the nunjber operator, multiplying by the level k. R e m a r k . Some of the Lie algebras of dimensions three and four correspond to various choices of the coefficients ijt, qk, dk- It turns out that our basic d a t a structures corresponding to classical orthogonal polynomials correspond as well to fundamental Lie algebras. The latter connection is a major theme of volume 1. In particular, the Poisson distribution, Poisson-Charher polynomials, corresponds to the oscillator algebra with ik = I, gk = t + k, dk = k, and the Gaussian distribution, Hermite polynomials, corresponds to the iJeisenberg algebra with ik = I, qk = 0, dk = k. The discussion here applies to the Meixner polynomials class in general, see Chapter 4 for complete details, but in fact we will apply it here explicitly for just the Poisson and Gaussian cases. Observe that
32
1.3.1 P r o p o s i t i o n .
CHAPTER 2
On the monic polynomials,
ipn{x) = io • • •
in-i4'n{x),
Thus, iPn ^R"oNow we define the polynomials:
coherent states
that are exponential generating functions for the
^-^ n! n=0
n=0
Note that this has the operator formulation •<pa = e " ' ^ o
1.3.2 P r o p o s i t i o n .
On the coherent states we have: -RV'a = -g- ^ a ,
l^'Aa = a ^Aa
Proof: These follow from the definition and the Proposition above describing the action on the tpn• From the general theory of Meixner polynomials, we have the following structure. Given that the moment generating function for Xt is
(e^^')=e'^(»)=^i^M^) n=0
Define the function V(s) as the derivative of the function Ho{s), the function H{3) for the centered distribution, i.e., with mean zero. And we denote the functional inverse to V by U, i.e., V(U(s)) = s. 1.3.3 P r o p o s i t i o n . the form
For the Meixner polynomials ^^ ^
class, we have the coherent state in
gXl/(a)-«H(J/(a))
where e*^^'' is the moment generating function of Xf. In for the Poisson distribution: log(l + v). for the Gaussian distribution:
particular,
H{s) = e' — 1, Ho{s) = e' — I — s, V(s) = e' — 1, U(v) = H(s) = 5^/2, with V(s) = s — U(s).
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
1.3.4 P r o p o s i t i o n . relation
33
For the Poisson and Gaussian cases, these functions H{a + 6) - H(a) - H{b) =
This is called the cocycle
satisfy
the
V{a)V{b)
identity
Notice that by orthogonality, the inner product of two coherent states is a generating function for the squared norms ||^„(p. However, using the observations above, this can be calculated directly as follows. 1.3.5 P r o p o s i t i o n . For the Poisson and Gaussian cases, the inner product coherent states is given by
Proof: We have (e"^'e''^') = (e(«+'')-^') = exp(tH{a identity, using the formulation of Prop. 1.3.3, V,„i, =
Coherent state
+ b)) and thus by the cocycle
gtlHiU(a)+U(b))-HiU{a))-H(U{b))]
^ ^tViU{a))V(Uib))
l.S.l
of the
^^tab
representations
Playing the role of a Fourier transform for this operator calculus is the CSR, coherent state representation. For an operator, Q, this is given by the normalized matrix elements (Q)a6-(V'a,W6)/(V'a,^6) Note that the adjoint Q* satisfies (Q*)ab ~ {Q)ba1.3.1.1 T h e o r e m .
and for the number
For the Poisson and Gaussian cases, we have for R and V
operator
The joint coherent state representation,
for the product,
is
(eP^ e»^ A^^)„6 = exp(apt + qb\ + a6t(A - 1))
34
CHAPTER 2
Proof:
We see that
so that the result for e*"^ follows by Prop. 1.3.5 after dividing out ^ai- For e ' ^ , use the relation Vipt = brj^i,. Thus, e*^ tpb = e'* V'ti and the result follows. The number operator acts as °° 6" RVtpb = y~l —rnrp„(x) °° ?)" ^—' n! n=0
Thus Hence the result. For the product, we have
Now apply Prop. 1.3.5. 1.3.1.2 Corollary.
•
We have the coherent state {R)ab=at,
representations
{V)ab = b
and {RV)ab = abt,
Proof:
{{RVf)ab
= aHh"" + abt
For example, ^
(e'"')ab = {Re'"')ab = ate""*
and evaluating at p = 0 yields the result for R. In general, differentiate with respect to the appropriate parameters and evaluate at 0 for R and V, 1 for RV, to get the result. • From the CSR's we see that the adjoint of R is tV, with CSR tb. R e m a r k . Note that the coherent state representation of A^^ is the same as the moment generating function for a Poisson distribution with parameter abt. Now for a main result. 1.3.1.3 T h e o r e m .
For an operator Q, we have
the CSR of Q with respect to the variables V{a),
V{b).
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS Proof:
35
From lAa = e^^(")-*«(^(«)), we have Vv(a) = e^"-*"*"). Thus, e
ax
= e
tH{a)
•
But we have via the cocycle identity ^tH{a)^tH(b)
^
g(H(a+t)g-(V(a)V(6) tHia+b) 11
1.4
OPERATOR CALCULUS AND DATA STRUCTURES: KNUTH'S MODEL
Now we apply this to the analysis of data structures. So, we are given a model for the data structures. Define the operators iZ, iV, L as above where i t , gjt, dk are the corresponding number of possibilities for the operations. In general, we denote the operator corresponding to the operation O by io- Thus ^/ = i i , ^Q = A^, and ^ D = L. For Knuth's model, ijt becomes t. And for dk = k, as for linear lists and dictionaries, we have L = tV. 1.4.1 L e m m a .
In Knuth's
model, the level crossing numbers
NOknit)
Proof:
=
satisfy
Y,{Xr'-Hok){X\<j>k)lek
In the formula NOknit)
= N p o s ( 0 , i : ) 5]iro,t,i(<)fl't',o,n-i-i(<)
substitute the formulas giving the path numbers in terms of matrix elements. What must be seen, then, is the significance of this formula. It really contains the assumptions of the model. Expanding in t, the superscripts denoting the number of insertions/negative queries:
^ - i V O L = Npos(0,fc)^ 53 s=a
'
—Hl,,Hl.^,^^_,_,
i a,6=0
Combining like powers of t on the right-hand side yields
Nou = Npos(o,fc) E E ( J Kk,Hir;^n-, This shows the counting of insertions/negative queries along the path, equally distributed • along the initial and final segments, with the break at the k^^ step. Observe that the cost COk of the operation on a file of size k can be expressed in terms of a cost operator, Ca{RV) such that Co{RV)i^k = COk 4'k-
36
CHAPTER 2 First main formula for integrated cost
1.4.2 T h e o r e m .
For an operation O and corresponding operator ^o
Proof:
Applying the cost operator to the level crossing formula given by the lemma
k
k
i
k
i
= Y.'^c'o{RV)CoX?-'-'x\)
= Y,{xr'-'ioCo{Rv)x\) i
where Parseval's formula is used to go from the second to the third line, 1.4.3 Corollary.
The total cost satisfies:
1.4.4 Corollary.
If the costs are all equed to 1, then we have Kn = n.
Proof:
•
We have ^ ^ (,o = Xt, thus
Kn{t)Hn=J2^Xr'^'XtXi) i
= n {X^) =
nunit)
Multiplying by e~* and integrating from 0 to oo, recalling that H„ — J^ e ^ ' /J-nit) dt, the result follows. • 1.4.5 T h e o r e m .
Second main formula for integrated cost
The ordinary generating function for KO„{t)Hn
satisfies
CO
^
s " KO„{t)H„
= ^((1 - sX)-^^oCo{RV)
(1 -
sX)-')
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
37
Proof: By the first main formula, writing Q for ^oCoiRV), the first term is zero, oo
noting that for n = 0,
oo
n=0
n=l i oo n=0
which gives the result.
i
•
A similar result holds for exponential generating functions. First we recall a basic analytic fact 1.4.6 L e m m a .
T i e convoJution formuJa s^_
n\ Proof:
r uP (s - M ) " - ^ - '
Jo p! {n-p-
1)!
Let u = st. Then the integral reduces to the beta function (Intro., §2).
•
Now we have the 1.4.7 T h e o r e m . The exponential
Third main formula for integrated cost
generating function for KOn(t)Hn
satisfies
equivalently, V-JtO„(0-ff„=e*«W n=0 "•
Proof:
/
{ioCo{RV))vi,-u),v(u)du
•'O
By the first main formula, writing Q for
n=0""'
^
^oCotRV)
'^
1
which gives the first form of the result. The second form follows by the theorem on CSRs, • Theorem 1.3.1.3. For Knuth's model KOn{t) the integrated costs.
must be multiplied by e~* and integrated from 0 to 00 to get
Along with this remark, to get the number of histories H„, we observe
38
CHAPTER 2
1.4.8 P r o p o s i t i o n .
Proof:
The generating function for the Hn is given by
Multiply by e * and integrate the relation °°
n
n=0
from 0 to oo. This yields the result, recalling that the moments convert to the H„. 1.5
INTEGRATED COSTS IN KNUTH'S MODEL
Now we apply the results of the previous section to find the behavior of the integrated costs. We consider the case where the cost function is linear in k. Thus, we consider the function COk = k. As noted above, the contribution of a constant to the integrated cost is of order n. We will see that the contribution of a cost of k is of order n^. Here we will find the leading asymptotic behavior of the integrated costs for list implementations. First, we consider linear lists and dictionaries, so that ^D = L = tV. With the cost function k, we need the coherent state representation {RRV)ab for insertions, for queries we need {RVRV)ab, and for deletions, (tVRV)ab- As in deriving the Corollary to Theorem 1.3.1.1, i.e., differentiating with respect to parameters to bring down needed factors, 1.5.1 P r o p o s i t i o n .
From Tieorera 1.3.1.1, we have {RRV)al> = a'^bt^ {tVRV)ab {RVRV}ab
^bt + abh^ =
a^bH''+abt
We see that the leading asymptotic behavior is given by the terms involving t^ as these correspond to the highest order singularities. 1.5.1 Linear lists This is the Gaussian case, so that V{s) = s. We will work an example in detail. Consider deletions at a cost of fe on a file of size k. From the above Proposition, then, from the CSR of tVRV we consider ab^t^. Using the third main formula, we compute Jo
12
Multiplying by e " ' and integrating from 0 to oo gives the generating function
Expanding in binomial series gives the result
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
1.5.1.1 T h e o r e m .
1.5.2
For linear lists we i a v e the asymptotic
39
behavior
Dictionaries
This is the Poisson case. Here H(s) = V(s) = e' — 1. First, 1.5.2.1 P r o p o s i t i o n .
The number of histories Hn satisGes H„
1
where I = log 2.
Proof: The generating function for Hn/nl is 1/(1 — H{s)). In this case, (2 — e')~^. Apply the technique of the Intro., §5.1. There is a first-order pole a,t s = I. To evaluate A, note that this is just the residue, which can be found by evaluating the derivative of the denominator at the singularity. This yields the factor 1/2 and the result follows. • Using the third main formula, we consider e"^'*' t^ times the coherent state integral. Multiplying by e~* and integrating from 0 to oo gives the generating function for A'„if„. Note the calculation, for s near 0, so that |Jy(5)| is small,
e-U^e'"^'Ut^ Jo " ~" -^ (1 - His)r As noted above, the pole is at / = log 2. To get the asymptotic behavior, we use the technique of the Intro., §5.1. So we calculate the coherent state integral and evaluate at s — log 2. Since the singularity is a 3rd-order pole, we get asymptotic behavior of order
The coherent state integrals and their corresponding contribution to the integrated cost are as follows: For negative queries, here we just multiply the number operator RV by t. Thus, using the Corollary to Theorem 1.3.1.1, after extracting the factor of t^, we have just ah or
v{s-u)v{uy. f (e" - l ) ( e ^ - " -l)du
Jo
= se' - 2{e' - 1) + s
Evaluating at s = I gives a contribution of 3/ — 2. For insertions and deletions: I f ( e " - " -l)du Jo
= e' (e' - 1) - 2se' + 3(e^ - 1) - ^ e ^ ' - 1) - s
40
CHAPTER 2
7 Evaluating at s = / gives a contribution of - — 5Z. For positive queries: f [(e" - l ) ( e ' ' - " - 1)]2 du = se^' + e^' - 1 + 3 - Ae' (e' - 1) + ise'
- 4(e" - 1)
Jo Evaluating at s = / gives a contribution of 13/ — 9. Combining these with the appropriate factors from Chapter 1, §2.2, we have these contributions: 19 For unsorted lists: (14/ ;r)/2 For sorted lists: (6/ - 4)/2 Using the Intro., Lemma 5.1.2, we see that we pick up a factor A = (1/2)^, and l / r ( 3 ) = 1/2. Combining these gives the behavior:
Scaling out the factors of H„, the net effect of the factors other than powers of / is a factor of 1/8. So, 1.5.2.2 T h e o r e m .
For dictionaries,
we have the behavior of the integrated
costs:
/^ 7 19 \ 2 , V For unsorted hsts: — — —— -—r n V 4 log 2 16(log2)2; , ,• . 3 1 For sorted hsts: 4 log 2 2(log2)2 1.5.3 Priority
queues
Here we use the functions 1/(1 — sx) in place of the exponential function. Thus, we apply Theorem 1.4.5. The polynomials that play a basic role here are the Tchebychev polynomials of the second kind, see Ch. 1, §6.2. 1.5.3.1 P r o p o s i t i o n .
We have the
expansion
1
°°
where M{s) is the moment generating
function
"(•) -
^
Vis) =
^ - ^ ^
and
DATA STRUCTURES AND ORTHCXJONAL POLYNOMIALS
41
Proof: This may be verified by substituting the indicated expression for V{s) into the generating function
i;«"^"(^) = r r 2 ^ For the appHcation at hand we note the 1.5.3.2 Corollary. have the expansion
For the polynomials
(t>k{x,t) = f^'I'^Ukix
12-^1) of Prop. 1.2.1.1, we
oo
—SX = Mis)Tvisr M^) 1 — n^ = 0^
— .QT
wijere M{s) is the corresponding moment generating ... . 1^^^^ =
function
Vl-is^t 2SH
and ^ '
2s
To apply Theorem 1.4.5, we need to expand the coefficients V(s)" in powers of s. This is where we use Lagrange inversion. 1.5.3.3 P r o p o s i t i o n .
Let V(s) = (1 - ^ 1 -As'^t)/2s.
Then
-w-S.'f(,„,V)'Proof:
Let x = V{s) = (1 - Vl - 'is'^t)/2s.
Then one readily finds that
X
s = X2 +t Applying the Lagrange inversion formula, Ch. 0, §5.3, we have, with XQ = SQ = 0, writing D for d/dx,
jt=i
Expanding (x'^ + <)* by the binomial theorem and differentiating accordingly, the result follows. • Now we will calculate the asymptotic behavior of the integrated cost for unsorted lists. Referring to Ch.l, §2.2, we see that deletion on a file of size k costs k — 1.
42
CHAPTER 2
1.5.3.4 L e m m a . For unsorted list implementation generating function for the integrated costs
E
of priority queues, we have for the
mV{s) 2m+3 Stm+2
m=l
Proof: 'We\ieixe(^oCo{RV)(j>„ — {n — l)^ri~iwe use the expansion of Prop. 1.5.3.1 to get
In the second main formula, T h . 1.4.5, oo
s{{l - sXr'ioCo{RV)il
- sX)-')
oo
= sM{sfi^V{srK = sM{sfV{s)
E
V ( 5 r ( m - l)^„_i)
Y^ X^(s)^'"m<-'"
using the squared norms ||^nlP = *""• Now observe that M ( s ) = V{s)/(st). Substituting in the formula above, the result follows. • 1.5.3.5 T h e o r e m .
For unsorted hsts, the integrated Kn ~
Proof:
cost for priority queues
satisfies
hny/im
In the Lemma, substitute in for V^'"+^(5) using Prop. 1.5.3.3 to get
Adjusting the index n leads to + l E ^ ^ " ^ E - ( 2 m + 3)(n 2n + rn + 2 We can rewrite the inner sum in the form
y:(k-2)i2k-i) ^0
^\+^ 'n-k
C^]
+ l\n + kj
Now we apply the central limit approximation, Ch. 0, §5.2, to get the asymptotic behavior of the inner sum, see Intro., Lemma 5.2.2, with A'^ = 2n, p = 1,
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
43
The initial factor of | comes from the fact that the sum is only for k > 0 so that only half of the Gaussian integral contributes. To get the integrated costs, we have to integrate out the factor of t", this gives
2n + l
V"/
Integrating the expression for the moments, from Prop. 1.2.1.1,
n+1 yields n\/(n + 1) x (")• follows. •
\n/
Dividing this into the above result, eq. (1.5.3.1), the theorem
For priority queues, we list the results for some other implementations. The behavior of K„ as n —+ oo is given by: Sorted list
Kn = j n^fim + 0{n)
Binary tournament Pagodas
Kn = | n l o g n + 0{n)
K-n = n log n + Oiri)
Observe that the pagoda provides the most efficient implementation for priority queues in Knuth's model. 1.6
ENUMERATION O F HISTORIES IN KNUTH'S MODEL: CONTINUED FRACTIONS
We review the technique of continued fractions to derive the generating functions for the number of histories. In the Introduction, §3.2, we indicated how the three-term recurrence for the approximants of the continued fraction gives a family of orthogonal polynomials, t/i„, the reciprocal polynomials of the denominator polynomials. The connection with orthogonal polynomials is based on classic work of Stieltjes and Markov. See Jones&Thron[47] for details (especially, pp. 241-256, 331-333, 342-244). Uaiko-v's Theorem says that the continued fraction, of the type in §3.2 of the Introduction, converges to the generating function for the moments of the distribution with respect to which the polynomials ip„ are orthogonal. Another way of stating the result is that the continued fraction (the corresponding J-fraction) converges to the Stieltjes transform of the distribution. Below, we will illustrate this in our context. We indicate the continued fraction approach to counting histories, detailed in Flajolet[23]. Given an alphabet A = {A,B,C,...}, a regular expression can be used to prescribe the form of words one is considering. Juxtaposition indicates concatenation, a -|- indicates 'or' and * is used to indicate 'arbitrary number of occurrences'. Thus {A+B)* indicates the language with alphabet {A,B}. Given a class of words, one can map a regular expression into a formal series, via a morphisra, with the letters of the alphabet mapping to variables.
44
CHAPTER 2
For example, words { A* }, each consisting of a string of A's of arbitrary length, maps to the geometric series
via the correspondence A <-> a. 1—a Knuth's model counts the number of ordered possibilities of i insertions as i\, regardless of the size of the file, as long as it has total length at least i. Thus, for dictionaries, since in the model considered here, insertions and negative queries are combined, the number of histories of a given length n, say, depends only on the total s = i + q, for i insertions and q negative queries. In the generating function, we denote by r the combined number of deletions and positive queries. Denote by H^ the number of histories of length n = s + r. We consider the generating function :
H{t,x,z)=
Y.
^^'-^"
i+q=s i-\-q+r—n
We denote by fff*! the corresponding generating function for histories of height < h. 1.6.1 T h e o r e m .
For dictionaries, H(t,x,z)
has the continued fraction
H{t,x, z) = l / l — t — qoz — d\xz
expansion
\ — t — q\z — d^xz / ...
where q^ = Npos (Q"*", A;), dk = Npos (Z?, A;). For linear lists and priority corresponding result holds, after setting qt = 0, k > 0, and t — 0.
Proof:
queues,
the
Using the method noted above, define the alphabet A={I,Q-,Qt,Qt,--,Di,D2,...}
where Ok, O € {D, Q'^}, denotes operation O performed on a file of size k. Let 5'''' denote the set of schemas represented by words having height < h, with initial and final level 0. The SW axe given by the following regular expressions;
5l°] = (Q- + Q+r S^'^ = (Q-+Q++IiQ-+QtrD^r 5t^l = {Q- + Qt + HQ- +Qt + I{Q- + Q^YD^YD^y In general, Sf*"*"^' is obtained by the substitution
Q- + Q+ + /(Q- + Qt+,YDH+^ ^Q-+Qt in S'- '. Via the morphism I <-y x,
Q~ «-> t,
Q'l *-t qkZ,
Dk <-^ dkz
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
45
we get I - t - qoz
ifW(<,x,z) = 1 and so on. Generally, W'^~^^'(t,x,z)
\ —t — qoz
d\xz 1 — t — qiz
is obtained by the substitution
dh+ixz : ^ t + qhz 1 - t - qh+iz in H^'^^t,x,^). Now let h go to infinity. The result for linear lists and priority queues follows from the fact that there are no queries. • t + qhz +
For priority queues and linear lists we can recognize the continued fractions directly. 1.6.1 Priority
queues
Here we have no queries, and dk = I for all k (Ch. 1, §2.3.2). Thus the continued fraction l / l
—x z / l
—x z / . . . / l
— xz/...
Call it 77. As this is a periodic continued fraction, substituting s'^ = xz^ we see that it satisfies 1 2„ =
1-5-^77
and thus 1 — T; + s'^rj'^ — 0. Since we want the solution that equals 1 at s = 0, 1 =
4s'2
l-Vl2s2
Comparing with the moment generating function connected with Tchebychev polynomials of the second kind given in Intro., §6.2, we see that this equals ^(25)^"/i2n- Comparing with Theorem 1.6.1 and using the formula for the moments, we have
1.6.2 Linear lists Here the continued fraction has the form \ I l-xz
I \-2xz
I ... I \-hxz
To identify this, consider the hypergeometric function Fia,b,s)=2Fo Writing out the series, one checks:
I ...
(1.6.2.1)
CHAPTER 2
46 1.6.2.1 P r o p o s i t i o n .
We have the
recurrence aSiFo
Write this, using the symmetry F{a,b,s) F{a,b+l,s) Fia,b,s)
= 1
= F{b,a,s), 1
• as
a + 1,6+1
as
F{b+l,a + l,s) F{b+l,a,s)
Iterating yields the continued fraction 1 / 1-as
/ l-{b
+ l)3 / l-{a
+ l)s / l-{b
+ 2)s /
With a = | , 6 = 0, 5 = 2xz, we recover eq. (1.6.2.1). Since F{a,0,s)
...
— 1, we have
oo
i^(|,l,2x0) = ^ ( | ) „ ( 2 x ^ r n=0
and hence /f3"„ = n ! 2 " ( i ) „ where we recognize the moments for the Gaussian distribution given in Intro., §6.2. Thus, we should expect a connection with the Hermite polynomials, as indeed we have seen. 1.7
DYNAMIC DATA STRUCTURES WITH FINITE POPULATION
So far, we have assumed that the universe of keys is infinite. Since in practice there are only a finite number of keys, it is of some interest to see this explicitly taken into account. Flajolet and Fran^on considered the markovian model and gave the time cost generating functions. Here we will follow Frangon, Randrianarimanana&;Schott[34]. We will state the results of their analysis, without presenting details here. The basic hypothesis are: i) the keys belong to a finite universe, the initial set of keys, [N] = { 1 , 2 , . . . , A''} ii) starting with FQ = 0, if at step i the set of keys in the structure is Fj C [N], then the universe at step i is [N] — (F, U Ui) where f7,- is a set of discarded Jceys, with Uo = 0. The operations are as follows : insertion: I{xi+i) is defined if 2;;+! € [N] — {Fi U Ui). F,+i =F,\J{x,+i},U,+i =Ui.
At the next step we have
deletion: £>(x,-|-i) is defined if x^+i € i*";. Theni^i+i = Fi — {xi^i},Ui+i
= f/iUfari+i}.
queries: Q'^{xi^i) is defined if a;;.).! € Fi and then i^i-fi = Fi, f i + i = Ui. Similarly, Q-ixi+i) is defined if Xi+i € [iV] - (F, U t/;) and then Fi+i = F,, Ui+i = U,\J{x,+i}. In other words, to perform an insertion / at step i a key is taken from the universe and added to the structure. The operation D consists of discarding a key from the structure. A positive query Q~^ modifies neither the universe nor the structure, while performing a negative query Q~ consists of discarding a key from the universe leaving the structure unmodified.
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
47
1.1.1 Number of possibilities of an operation Here we give the number possibilities for each operation for the various data structures. This effectively defines the model. for linear lists: Npos(«*''I) = i V - i + 1,
0 < i < A^
Npos {D,k) = k,
0
for dictionaries: Npos (i*''I or Q~) = N - i + 1,
0
Npos(Z),fc) = A:,
0 < fc < iV
Npos{Q+,k)
= k,
0
for priority queues: Npos (i*''J) = N -t + 1, Npos{D,k)
= l,
0
for symbol tables: Npos (i*''/) = N ~i + l,
1.7.1.1 P r o p o s i t i o n . of levels (ko.ki,..
.,kn),
0
Npos(£),A:) = 1,
0
l>(pos{Q+,k) = k,
0
< N
For a schema O1O2 ... On which has a corresponding the number of corresponding
sequence
histories is
jJ^l[Npos(D,k,)l[NposiQ+,k,) ^ '' jes lep where S (resp. P) is tije set of indices j such that Oj = D (resp. Q'^) and i denotes total number of insertions I and negative queries Q~ in the schema.
the
The analysis can now be done with the same kind of techniques as for the infinite-keys case. 1.7.Z Continued fraction theorem for a finite universe of keys Let H^Xl+r t>e the number of histories of length n = p + q + r with p insertions, q negative queries, r = r\ -{- r^ where ri (resp.r2) is the number of deletions (resp. positive queries). Consider the following generating function :
^(M,.) = E<.v^^^^^^^-w iV! As before, we have a continued fraction expansion for H{t,x,z). ks may be expected, the expansions are similar to those obtained for an infinite universe for histories of height bounded hy h = N. For the particular data structures of interest:
48
CHAPTER 2
a) Dictionary H^{t,x,
z) = 1 A -t-Oz-
1 x 2 / l - t - l z - Ixzj
...
l l - t - N z
b) Linear list H f i ( 0 , x,z)=lll-
Ixzll
- Ixzl...
Il
c) Priority queue H ^ Q ( 0 , a;, z) = 1 A - xzll
- a;2 / ...
I\ -
xz
the fraction being of length N. 1.7.S Counting
histories
One can calculate the number of histories starting and ending with an empty file. The results are as follows: a) Dictionary i
where the .s„^,- are the Stirling numbers of the second kind. b) Linear list
c) Priority queue N_^.r^ (2n)'H2"n = (N)n
{n + iy.
1.7.4 Integrated costs: results One can find as well the integrated costs as a function of both n and TV. Here we state the results for K.^^. a) Priority queue - sorted list representation n'-y/nTT + 0(n),
n < N
-unsorted list representation -\/rMr + 0 ( " ) , binary tournament
n
DATA STRUCTURES AND 0RTHCX30NAL POLYNOMIALS
49
3
-n\og(n}
+ 0{n),
n < N
-pagoda nlog{n) + 0{n),
n
b) Linear list - sorted list r^ + hn 3 '
n<
N
- unsorted list n
+5n 6 '
n<
N
c) Dictionary - sorted list
-^^+(| + ^)^ + i+0-^)"(^^+(i-^)^)'
-^^
- unsorted list
+ ^'-N for n<
U^^ + ( " - T ) ^ - " + 2
N.
R e m a r k . Letting N —y oo recovers the results proved for the infinite universe case. We leave this for the reader. II.
M u t u a l exclusion
We have a system of processes which axe accessing a common set of resources. Each process either accesses or releases a resource in a given time period. The system is ruled by the restriction that only one process may access any given resource at a given time and (consequently) each process releases at most one resource. The objective is to avoid a deadlock state where processes have no available resources. There are various ways to model this. One way is to use a combinatorial approach, which will be discussed in Chapter 5. Another approach is to use a probabilistic model, e.g., a type of birth-and-death process
50
CHAPTER 2
counting the number of active processes going on and assigning a certain probability of the system locking up. Consider p processes PI,P2T - • ,Pp and r resources Ri,R2, •. • ,Rr- A process P, may access resource Rj, denoted by Pi(Rj), or it may release it, denoting this by Pi{Rj). Fran5on[31] considers all 'behaviors' corresponding to a given resource Rj. This is the language C{Rj) which we represent by the formal sum:
k>0
l
(assuming that all resources have been released at the end). Since r resources are used, the behavior of the full system is an element of the shuffle product: LlJi< j < r £{Rj). The idea is now to use the exponential generating function. This is effective since the exponential generating function of the shufHe product of two languages associated to disjoint alphabets is the product of the corresponding generating functions. By this method, Frangon proves that the exponential generating function of the admissible behaviors is: F{z) = c o s h ' " ( v ^ ) This leads to the asymptotic result for the number of admissible behaviors
&s n —* oo. We present an alternative probabilistic approach to the average behavior of such systems. R e m a r k . Raynal[75] is of particular interest for this topic. Also see Lavault[55]. The mutual exclusion protocol can be represented by a graph whose nodes correspond to processes at work, the edges indicating requests for resources. For example, P I —> P2i—
P 3 <— P 4
P6—>P5 means that the processes P I £ind P3 wait for some resources used by P2, while P 4 (resp. P6) waits for a resource used by P 3 (resp. P5). Assume that P2 releases the resource requested by P 3 , we get the new graph P I —^ P 2
P 3 <— P 4
P6—>P5 Assume that P3 asks for a resource used by P6, we get: P I —• P2
P S <— P 4
i P6 — » P 5
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
51
Should P3 ask for a resource used by P I , the system runs into a (partial) deadlock situation. Another view is this. If the state Ek indicates that k processes are at work with the mutual exclusion protocol, we have the following graph of possible transitions: Ek-i
<— Ek —*• Ek+t
i D the state D corresponding to deadlock. 2.1
FELLER-KOLMOGOROV'S EVOLUTION EQUATION
At this point we may model the situation as a birth-and-death process modified by the addition of a deadlock state. Assume that the probability of transition over the time interval [t,t + At] is exponentially distributed. Denote by Pk{t) the probability that k processes out of n are at work at time t. We assume the transition probabilities are given according to: P{Ek-i -> Ek-, [t,t + At]) = XkAt + o(At) P(Ek+i
-^ Ek] [t,t + At]) = fikAt + oiAt)
P{Ek -^ D; [t, t + At]) = akAt + o(At) We have then: Pk{t + At) - Pk{t) = {-{ak
+ \k + tJ.k)Pk{t) + \kPk-i{t)
+ / i t P t + i ( i ) ) At + o(At)
Dividing by At and letting At —> 0, we have the system of differential equations; Pl{t) = ~iak + \k + ^^k)Pk{t) + XkPk-iit)
+
fikPk+iit)
with boundary conditions : Pi'(t) = -{ai
+ fi.i)Piit) +
Pl,{t) = -{an
+ )ln)Pn{t)
fiiPiit)
+ A„Pn_i(t)
Writing P ( t ) as the row vector with components Pk{t), we can write the system in the form P'(t) = P ( t ) M where the matrix M has columns ( . . . , A i t , - ( Q t + \k +
fik),fik,---y
Note that the ajt are the infinitesimal rates at which the system deadlocks. See Feller[22] for thorough background on Markov processes and how to analyze such systems.
52
CHAPTER2
2.2 APPROACH VIA RECURRENCES Here we can apply the transition approach of Section 1 as well. Assuming that access/release of a resource is independent of the time, n, and that access/release depends only on the number of active processors, we have a recurrence of the form Hk,n+1 = dk+-iHk+l,n
+ qkHk,n
+
ik-lHk-l,n
where Hk,n is the number of histories such that k processors are active at time n. A deadlock is modelled by the fact that a path touches the a;-axis, i.e., there are no processors active. To calculate the behaviors, we could enumerate all histories whose height is strictly greater than 0. This is like the problems studied in Section 1 and similar methods are applicable. Karlin-MacGregor[48][49] have studied the relationship between birth-and-death processes and orthogonal polynomials. For studies from the point of view of concurrency measures, see Arques, Frangon, Guichet&Guichet[6] and Geniet, Schott&Thimonier[37]. III.
E l e m e n t s of duality t h e o r y
Here we briefly indicate the general approach to duality — the correspondence between the recurrence for the c„jt and the recurrence for orthogonal polynomial basis vectors used in §1.1. The idea is to interpret Cnk as components of a vector C „ . You have a discrete dynamical system of the form C „ + i = ACn with C „ a vector of components Cn(k). operator X is determined by the relation
This gives a matrix C{n,k)
k
where $ is a column of the basis vectors j. Thus
And X " + V o = X{X"4>o) = C „ • X $ = ACn • * = C „ • A*$ So the action of X on $ is dual to that of J4 on C: X^
= A*^
— C„(/;).
The
DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS
53
Example. Consider the factorial powers x^"' = x{x — l)(x — 2) • • • (x — n + 1). Then the Stirhng numbers of the second kind are determined by the relations C" = y^^Sn^kX k
With 4)k = x'*', we have the action of X X(f>k = ix-~k
+ k} x<*^ = k+i + k(f>k
By duality, this gives the recurrence for the Stirling numbers Sn+l,k
= Sn.k-l
+ ^•Sn.lt
R e m a r k . Basic studies concerning Knuth's model are Frangon, Randriajiarimanana&Schott[33] and Randrianaximanaj:ia[74]. The probabilistic approach is taken in LouchEird[58], and Louchard, Randrianarimanana&Schott [60]. See Maier[63] as well. A basic reference for this chapter is Flajolet, Pran5on&Vuillemin[26]. Also see Flajolet[24], and Fran5on[30]. For various applications in combinatorics via an approach similar in spirit to that taken here see Godsil[38]. Meixner discusses the polynomials now known as those of Meixner type in Meixner[69]. A presentation of finite operator calculus is given in Rota[78].
Chapter 3 A P P L I C A T I O N S O F BESSEL F U N C T I O N S A N D LOMMEL POLYNOMIALS
In this chapter we present some applications of Bessel functions. Section 1 is devoted to the symbol table in Knuth's model while, as a complement, section 2 discusses the rate of convergence of zeros of Lommel polynomials to those of the corresponding Bessel function. Section 3 concerns the analysis of some concurrent algorithms, particularly some aspects involving special functions. I.
A n a l y s i s of t h e s y m b o l table in K n u t h ' s m o d e l
The approach developed by Flajolet, et al., leads in this case to a divergent generating function. So a different approach was called for. Recall that for Knuth's model, one considers the generating function Hn(t) = Yli'^'/^^•)-^n where s counts the number of insertions. A study of the numbers H „ ( l ) as a first step towards the solution was made in Flajolet-Schott[27] where the asymptotic behaviour of these numbers was obtained with the help of Bessel functions. We recall these results in the next section. 1.1
BESSEL NUMBERS
Here the notation H„ refers to iif„(l), i.e., the function H„(t) evaluated at t = 1. Flajolet&Schott[27] proved that:
Hn^Y.
n+2
as n —> oo
lt>l
/f„Ri(n/2elogn)" R e m a r k . The numbers H„ are denoted 5 * in [27]. The starting point for the study is the fundamental recurrence for Bessel functions : J^+i{x) = 2vx~^ Jv{x) —
Jy-i{x)
Rewriting this relation as: ^-J^^
= 2vx-'
-ll{Ux)IJ^+,{x))
and iterating the process, setting x = 2, yields
This continued fraction provides the connection with symbol table histories for the analysis of this data structure. We now recall some of the results from Flajolet-Schott[27].
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
1.1.1 P r o p o s i t i o n . The asymptotic Bessel functions is expressed by
expansion, as u —* oo, of a quotient of consecutive
vj,{i)
1.1.2 L e m m a .
55
Z ^ " ' " ' " pn+2 n>0
We have J.-i(2) vJ^{2)
1 1V
"'•
r>0 ^
'''•
where
and the Bessel numbers Hn
satisfy
The asymptotic behavior of the H„ axe deduced from these results. 1.2
SYMBOL TABLE AND BESSEL FUNCTIONS
For the symbol table in Knuth's model we have the recurrence, cf. Ch. 2, Prop. 1.1.1.3, X(f>k = <Jt+i + k(i>k + (f>k-\
with squared norms ek{t) = ||^jt|P = ^""*1.2.1 P r o p o s i t i o n .
The solution to the
recurrence
X(j>k = t(j>k+l + k^k + k-l with -\ = 0, 1^0 = 1 may be expressed in terms of Lommel
polynomials:
4>k = r * / 2 i ^ , , _ , ( - 2 ^ / < )
Proof:
Observe that (l>k = t'~''''^ipk{x/\/i)
where the polynomials V"* satisfy
xipk = ^k+i + kt~^l'^il)k + V"*-! Comparing with Intro., eq. (4.1.5), yields, with e — -y/t, ^k = i i t , _ , ^ ( - 2 v ^ ) from which the result follows.
•
R e m a r k . In the following, we will denote the argument —1\fi of the Bessel function by r . An important feature comes via Intro., eq. (4.1.6).
56
CHAPTER 3
1.2.2 Definition. is denoted by E^.
For a given r , the set of zeros of the function of x given by J _ i _ i ( r )
Now we have 1.2.3 L e m m a .
Let (^ € E^ be a zero of J-I^X{T)-
Rn.-dr)
Then
J-C(r)
Proof: Replacing in Intro., eq. (4.1.6), ex by x and the argument —26 by r , we have, dividing out J-^iT), ^"•-^^^) -
and hence the result.
- j l A ^
+
J_.(r)
^"-1-1-HO
•
It turns out that t h e set S r will correspond to the values of x where the measure of orthogonality for the (j>i: is concentrated. From the recurrence relation we see that one thinks of the values of x as eigenvalues of the matrix operator given according to the right-hand side, hence we have the spectrum S^. To see this, we use the continued fraction, cf. Intro., §3.2, which gives the moment generating function, M{3) = ((1 — sX)~^), as we saw in Chapter 2. Here we have M{s) = i l l - t s ^ l - s - ts'^/l
-2s-
ts'^/l
-3s-
ts'^f • • •
corresponding to the monic polynomials associated to the (^t, satisfying the recurrence xtpk = ^k+i + krpk + i^k-i, as noted in the remarks preceding Prop. 1.1.4 in Chapter 2. In particular, both sets of polynomials have the same measure of orthogonality. The Bessel numbers of the previous section show that as a power series at zero, it has radius of convergence equal to 0. However, the continued fraction gives the appropriate analytic continuation to regions corresponding to existence of the Stieltjes transform, which is related by the transformation s~^M{s~^). 1.2.4 T h e o r e m .
The moment generating function has the formal
Mis) = ( - ^ ) = I-SX with T = —2y/i. The Stieltjes transform is
^
differentiation
^
sVtJ-l-l/s{T)
^s-X' ViJ-i-,{T) giving the spectrum S^ as the zeros of the denominator. measure is given by the residues
the prime indicating
^
expression
with respect to s.
The corresponding
probability-
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
57
Proof: From Watson [84], pp. 153, 303, as noted in the previous section, the ratio of Bessel functions can be expressed as a continued fraction which yields the equation
^ - ^ = 1/—+ 1 Z
Ji,\Z)
'
1/—+2
Z
Z
I
1/—+3
Z
Z
I
Z
1 Z
And thus, •
2 / 1
„
±2
f-\
J^^
o„
/i
A,*!
o_
1 - — = ts^ / I - 3 - ts^ 11 - 23 - ts' 11 - :ia - ts Dividing numerators and denominators by s y i we have
Setting z — —2\/t, v = —1/s, we have
M~
2v \ z
J^{z)
We get the Stieltjes transform, {{s — X)^^),
) ~
2v
via s~^M{ljs).
J^{z) •
R e m a r k . Observe that the Bessel function in the denominator satisfies
(-y^)-._,_,(.) = X : ; ; T r ^ SO the spectrum is given by the zeros of the function
E
n=0
^
'
Now we see that the discussion in [27] is for the case t = \. For general t we have the asymptotic form of the zeros / •
^"^"
< "
n\{n + \)\
as n —• oo. 1.3
INTEGRATED COST F O R T H E SYMBOL TABLE
We follow the technique used in Chapter 2 for priority queues. We want an expansion of (1 — sx)^^ in terms of the polynomials (/>„. Write ( ^ - I ^ ^ n W ) = M{s)V„is)
(1.3.1)
58
CHAPTERS
We know that (t>o = 1, 0 by orthogonality. Introduce the operator A j acting on functions f(s) by A,/(s) = The principal feature of this operator is that A , ( l - sx)~^ = x{l -
sxy^
that is, the functions (1 — sx)~^ are eigenfunctions of A, with spectrum x. Applying As to both sides of eq. (1.3,1), it follows via the recurrence for 4>n that, for n > 0,
Applying eq. (1.3.1) to the left side of this equation yields M{s)
{tVn+l{s)
+ nV„{s)
+
Vn-x{s))
Thus, Vn{s) satisfies the same recursion as does i^„ except for initial conditions. Setting n = 0 gives, using 4>i — x/t, ,
1
y,
M{S)-1
via the operator As, while from the right-hand side we have
Therefore
From this one guesses the general solution
K(.) = * - " / ^ ^ ^ ^ i ^ cf. Intro., (4.1.6), which is readily verified. We thus have 1.3.1 L e m m a .
The expansion in Lommel __L_ -
•'•
\^.^n/2j.
polynomials /^\
Jn-l/sJT)
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS Proof:
59
From the above, we have the expansion 1
°°
J — ^
= Mis)
J2
Vn{s)M^)/£n
with the squared norins Cn = * - " . Now,
s\/iJ-i-i/^{T)
and hence the result.
J-I/,(T)
•
Now consider positive queries at cost fc on a file of size k. We have ^oCoiRV)4>n
= n^4>n
Set s^/i
J_i_i/s(r)
so that (1 — sx)~^ = J2 ^n(s)4>n(3;). Using the second main formula for integrated cost, Ch.2, Theorem 1.4.5, we want to calculate oo
3{{i - sXYHoCo{RV){\ - sxy^) = s J2 Wmisfmh-"^ using the orthogonality relations of the <(>„• Substituting back in the expression in terms of Bessel functions for Wm(s) yields oo
'>m-\ja\J)
-T St
J-\-\IS{T) ^
with Er the poles of the summands. So we look at an expansion in I/5 of
E
Jm-sir) \.J~x-.{r)
expanding in partial fractions the main contributions come from the terms with double poles: / Jm-s{T)
\
_
• ^
Jm-
y(s-cy
+ lower order
The Stieltjes transform says that
(7:^)=^ ''~^'
p(C)
C^.'-^
_
J-S{T)
ViJ-i-.(r)
60
CHAPTERS
with
Hence 1 Y^
Jm-<:{r) __ -s-^ Jm-dr? __ Y- McmO!^. 2
since, on E r ,
So, expanding the terms [s — C.Y, we consider
n
m
^
Now, Yl <J^m(C)^P(C) sums to ||
E^E-'(^>(^))n
m
with ( ) denoting expectation with respect to the weight function miC)'^PiC)t'"- So to get the asymptotic behavior, we find the value Cmax that maximizes 4'm{Q^p(0- This will give nH„{t)
X^mVCmax)'?im(Cmax)^i'" 771
as upper estimate for KOn{t)Hn- Dividing out by the Hn will yield the result. Though the calculations have not been completed at this time, preliminary computations (Louchard, private communication) indicate that the result is
(logn)i/^
1.4
RATE O F C O N V E R G E N C E O F ZEROS O F LOMMEL POLYNOMIALS
In this section, based on Feinsilver-Schott[20], we discuss the rate of convergence of the zeros of Lommel polynomials to the zeros of the corresponding Bessel function (considered as function of the order). The study was motivated by the symbol table analysis. The Maple program, whose results are reported at the end, illustrates that our result is quite good. J. Coulomb[13] first investigated the zeros of the Bessel functions Jv{x), with x fixed. He observed that the zeros converge to the negative integers. The study Flajolet-Schott[27]
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
61
gave a quajititative version of Coulomb's result in the context of some combinatorial problems related to non-overlapping partitions. Their work led to the question of the speed of convergence of zeros of Lommel polynomials to zeros of corresponding Bessel functions. It is likely that the technique used here will prove useful in similar cases for other classes of functions. In the following, the argument of the Bessel function is fixed and v is variable, and will be denoted by x. Consider the (modified) Lommel polynomials: [m/2]
.
.
n)„ r»=0 ^
^
with {x)a = x{x -{- l)...(x -\- a— 1). Throughout the following, x is any complex number that is not an integer. We have, in the notation of Watson[84], p. 294ff, Hm[x) =
R^,^{2)
We let
0 For m —> CO, recalling that x is non-integral, J{x + m) =
1 r(a; -l-m-l- 1)
11 + 0
and
(1.4.1)
J(—x — m) =
1
r(-x-m
+ l) 1 + < ^
where the O terms are uniform (in a;) on compact sets. The principal formula required is the following, which may be found in Watson[84, p. 295],
Hmix) =
[J{x + m)J(l
-x)
+ ( - ! ) " • J ( - x - Tn)J{x - 1)]
(1.4.2)
Now let /x be a positive integer. Define the domain fi^ = { x: 0 < |x -|- /i| < 1 }. The first lemma is related to Hurwitz' result for Lommel polynomials [84, p. 302]. 1.4.1 L e m m a . ('normally').
On il^,
r(H-m) converges to J(x — 1) uniformly
on compact
subsets
CHAPTERS
62 Proof:
By eq. (1.4.2),
o
1 r{x + m)T{x + m + 1)
7r
+ sinTTX
(-l)'"J(-a:-m)J(x-l) r(x + m)
The first term is negligible. We have to show that the coefficient of J{x — 1) in the second term converges normally to 1. In fact TT ( - l ) " ' J ( - x - m ) _ 71^ sinTra; r(x + m) sin7r(a; + p) T{x + n) r ( l -X - fi) — m) r(x + fj,)T{i-X-fj.) r(x + m) r ( i
l+O
where we note that the O terms are uniform on compact subsets of fi^. By the reflection formula for the gamma function, the first two factors reduce, and thus the expression is asymptotic to +^ Fjx + fi)Til - X - li) ^ ^ ' T{x + m)T{l - X - m) Recalling that the zeros of J{x) are very closely negative integers as a; —> — oo we have 1.4.2 L e m m a . Let J{( — 1) = 0. Let Cm be the zero of Hm(x) closest to (. Then, as oo, we have the asymptotic reJation
\u-c\~\j'ic-i)r
HmiO
r(c + m)
Proof: (Note; the idea is similar to that of Newton's method for solving equations.) Let fni{x) — p / ^ ^ N . So fm and Hm have the same roots near (. By the mean value theorem, there exists ^mi e.g. on the segment between ^ and Cm, such that fmiO-fmi(:m)
=
(C-Cm)f'miU)
Thus, since fmiCm) = 0, &s m —t oo,
IC-Cm| =
fmiO ImK^rn)
\fln(0\-' \fm(0\
where, since {fmix)} converges normally to J(x — 1), (by Cauchy's formula) the derivatives are equicontinuous. And for large m, we replace /'„{() by J'{( — 1). • And finally we have:
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
63
1.4.3 T h e o r e m . Let (^ denote a zero of J{x — 1). Let (rn denote the zero of Hm{x) nearest to (. Then, as m ~* oo,
Km - C I
Proof:
1 TT J(l-C) sin nC J'(( - 1) r(C + m)r(C + m + 1)
(1.4.3)
We apply eq. (1.4.2) at a; = (, so that the J{x — 1) term vanishes. Thus, ^m(C) r(C+m)
- j(i_c)^g±4) ' ' r ( C + m)
sinTfC
Combining eq. (1.4.1) with the above Lemma yields the result. R e m a r k . Relation (1.4.3) may be written in the form
ICm-CI
7(1-C)
r(-C-m)
j ' ( C - i ) r(c + m)
via the reflection formula for the gamma function.
64
CHAPTERS
Numerical Results
m 4 5 6 7 8 9 10 11 12 13 14 15
r-61Practice
Theory 1 X 10-^ 5 X 10-5 1 X 10-« 3 X 10-* 4 X IQ-"
9 X IQ-'' 4 X 10-5 1 X 10-« 2 X 10-* 4 X 10-^°
1^^-61
Theory
8 7 3 1 3 5
X 10-^ X 10-^ X 10-* X 10-= X 10-'' X 10-^
^, denotes a zero of J{v) = J„(2) (^ denotes a zero of a Lommel polynomial We have (approximately) ^^ = 0.2538058170, 6
Practice
4 4 2 8 2 4
r-^si Practice
Theory
X 10-^ X 10-3 X 10-* X 10-« 8 X 10-^ X 10-'' 7 X 10-3 X 10-"* 3 X 10-" 1 X 10-5 3 X 10-'' 5 X 10-^
4 4 2 8 2
X 10-2 X 10-3 X lO-" X 10-^ X 10-''
-4.9997743198, ^a = -7.9999999961
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
II.
65
Analysis of s o m e concurrent a l g o r i t h m s
The purpose of this discussion is to convince the reader that even very simple distributed algorithms can lead to difficult mathematical analysis, involving special functions and advanced probabilistic techniques. Problem 1: Consider a contiguous block of m locations, which we use to implement two stacks. One stack grows from the left end of the block and the other from the right end until the cumulative size of the stacks exhausts the available storage. This strategy is to be compared to a simpler one, namely, allocating separate zones of size m / 2 to each stack. This problem of Knuth has been investigated by Yao[85], Flajolet[25], and more recently by Louchard-Schott[61], Louchard et al. [59], Maier[64], and Maier-Schott[65]. Problem 2: Consider two customers P j , P2 sharing a fixed quantity of a given resource R. There are fixed upper bounds m i , 7712 on how much of the resource each of the customers will need at any time. The deadlock avoidance policy, known as the banker's aJgoritJim, affords to customer P,, (i = 1, 2) the required units only if the remaining units are sufficient in order to fulfill the requirements of Pj,{j = l , 2 , j ^ i). For a complete description of the banker's algorithm see Habermann[44], and Peterson&Silberschatz[72]. The present discussion is based on Louchard, Schott, ToUey, and Zimmerman[62], and Louchard-Schott[61]. See also [51]. 2.1
A PROBABILISTIC MODEL F O R T H E ANALYSIS O F T H E T W O STACKS P R O B L E M
The two stacks begin on opposite ends of the block of size m and grow until the cumulative size exhausts the available storage. The time to absorption and the final stack sizes are random variables whose distributions will depend on m and on the probabilities of the elementary operations : insertion 7, (resp. deletion £);), i = 1,2 is performed with probability p; (resp. qi). Obviously Pi + q\ + P2 + <12 = 1^ Therefore, as shown in Flajolet[25], the natural formulation of this shared storage allocation algorithm is in terms of random walks inside a triangle on a 2-dimensional lattice: a state is a pair formed by giving the size of the two stacks. Consider a particle that starts at the point (a, h) of the plane lattice and moves according to the transition rule {x + 1, t/), with probability pi {x — l,y), with probability 5i {x,y) {x,y + 1), with probability p2 , {x,y — 1), with probability 52 where pi + p2 + 5i + 92 = 1 It stops when it hits the absorbing barrier x + y = m. If the parameter m is the memory size, then we can say that the transitions above correspond respectively to random insertion / i and deletion Di (resp. /2, D2) in stack 1 (resp. stack 2). So a natural formulation of the two stacks problem is in terms of random walks inside a 2-dimensional lattice triangle with two reflecting barriers along the axis — as a deletion has no effect on an empty stack — and one absorbing barrier along the diagonal — the algorithm stops when the combined sizes of the stacks exhausts the available storage rn. (See Figure at the end of this section.)
66
CHAPTER 3
2.1.1 The banker's
algorUhm
In a similar way, we can see that the formulation of this problem is in terms of random walks inside a 2-dimensional lattice rectangle with a broken corner, four reflecting barriers and one absorbing barrier. (See Figure at end of this section.) 2.2
SOME RESULTS
The problem is to find the time and the position of the paxticle when it reaches the absorbing boundary. The general case involves four parameters, pi +P2 + 9i + 92 = 1; it is rather complicated and some problems remain open. Therefore we will restrict our study to the isotropic case, where Pi = P2 = 9i = 92 = 1/4. We refer the reader to the papers mentioned above for further details. Let Ym be the random walk described above with initial condition Fm(0) = Xm, and we denote by Zm (resp. T ^ ) the hitting place (time) on the absorbing boundary. Flajolet has given an analytical proof based on continued fractions and orthogonal polynomials, methods similar to those used for his analysis of dynamic data structures. Here we follow Louchard, who derives a diffusion approximation. The densities involved may be expressed in terms of theta functions, see Intro., §4.2. 2.2.1 T h e o r e m .
The asymptotic
distribution
ofYm{-)
is given by the foJJowing
M?!!!M ^ Wit) m
where W{t) is a two-dimensional (reHected and absorbed) Brownian motion (B.M.) with appropriate boundary conditions. The convergence is in the Skorohod sense. Let the hitting time tor W{-) be T. The density o{W{-) is given by T'AW{t)&dy,t
=
< S V ] exp{~k^7r'^t/2)cos{kT:xi)[cos(kTryi) *-
— cos[(A;7r(l — 5/2)]]
k=i 00
+ 2 N ^ exp( —A^7r^i/2)cos(A7ra;2)[cos(A7rj/2) — cos[A7r(l — yi)]] 00
+4 ^ k=l
00
^
exp[-(fc^ + A^)7r^
• [cos(fc7ryi)cos(A7rt/) - cos[A;7r(l - 2/2)] cos[A7r(l - yi)]] I dyidy2
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
67
Proof: We include a sketch of the proof. The proof presented here is based on ChungWilliams[ll]. The weak convergence is easily deduced from Chung-Williams, Th.8.4. Our reflecting probabilities are such that (in one dimension, for instance) Ymin) = Xin)
+ max ( - X ( i ) ) 0
where 1=1
and P(A. = l ) = p ,
FiAi
= -l)
= q
By the central limit theorem, for each t > 0, as n —^ oo, X" = {ay/n)~^X^„t] —> Bt in distribution with B a standard Brownian motion on R . For each m € N , {X"} converges weakly in D[0,772] to {-B(, 0 < t < m } . Then it follows by the continuous mapping theorem that X(" + max i-X^), 0
converges in the weak sense to Bt + max (-Bs),
0
Q
Since Bo = 0, we have max(—B.) = max BI 0<3
0<3
Weak convergence on D[0, m] for each m € N implies convergence in distribution for each time t, hence X r + max ( - X ; ) -^Bt+ max ( - 5 7 ) o<s
0
in distribution for each t. The continuous mapping technique used in Chung-Williams leads to the reflected B.M. W. Also the hitting time T is a continuous functional of W (almost surely) so that the weak convergence is still valid for our absorbed process. The classical one-dimensional B.M. with reflecting boundaries at 0 and 1 is well known, see Feller[22, Ex. X.5.e. and prob. XIX.9.11]. Its density is given by, calling this process Wi: Fi,[Wi{t)
£ dyi] = I l - | - 2 2^exp[-/i;''7r''V2]cos(fc7rxi)cos(/i;7r2/i) V dyi I. k=l = [l-^Ei(xi,yi)]c?i/i
say, and similarly for W2 • By a classical reflection principle ax;ross the absorbing boundary, we see that
F,[W{t)edy,t
=
| [ 1 -f- S i ( x i , y i ) ] [ l -I- 5:2(12,2/2)] - [1 + S i ( a r i , l - i/2)][l + £2(^2,1 - t / i ) ] | dyidy2
CHAPTERS
68 hence the result.
•
We now have all necessary ingredients to derive the limiting hitting distributions. The coordinates z in the statement of the Theorem are given by an appropriate translation and rotation, see the first lines of the proof below. 2.2.2 T h e o r e m . For the distributions W on the diagonal we have: P^[T€dt,ZiT)edz2]
r °° *• 4 = 1
of the hitting place (Z) and hitting time (T) of =
exp(—A;'^7r^t/2)cos(fc7rxi)—pr sin[A;7r(- -\—j=)] yl 2 -\/2
+ ^exp(-A^7r^V2)cos(A7rz2)^^sin[A7r(- -
-j=)\
A=l 00
00
+ 2_, y ^ exp[—(fc^ + A^)7r^
...
,1
Z2 , ,
r,
/ 1
-^2 1
x-=[/:sm[fc7r(= ) ] cos[A7r(=] ^2 + -V5 2 - -y/2' V2' +Acos[fc7r(l + 4 ) ] s i n [ A , r ( i _ i | ) ] ] | dtdzo
'2
v^
T i e marginal densities are given by P4Z{T)edz2] (
4
8
=
^cos(/c7rxi)
. r, , 1 , -^2^,1 , ^
cos(A7rx2) . ,.
,1
^2 .,
Y ^ V ^ cos(fc7ra;i)cos(A7rar2), ^^,^^,^ 4A:sinM^ + ^ ) ] c o s [ A . ( i - ^ ) ]
+ 1775 T T V ^ 2^ 2^" /t=l
A=l
+ Acos[^7r(- + •^]sin[A7r(- - - ^ ) ] ] | d z 2 and
p,[Tedt] •^4 ^ '^
= exp(-A:^7r^i/2)cos(fc7rxi)+ 4 ^
kodd>0
8
Y^
exp(-A^7r^
Xodd>0
E
e x p [ - ( P + A2)7r2
iorfd>0 Aet;en>0
-8
^ keven>0
J2 Xodd>0
exp[-(P
+ X^)7rH/2]{P +
X'')/(k^-\^)cos(knxi)cos{X7VX2)\dt
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS Proof:
69
Determine the new coordinate system (^1,22) by
yi = 2 + ( « i + ^ 2 ) / \ / 2
or
22 = -yiyi
-y2)
In this new system, let
^.{t,zuZ2)
:= P4W(t) G dz,t
It is well-known that the hitting density is given by F4T
€ dt,Z{t)
e dz2] = --[a.,v'x(<,^i,22)]z,=o
(this is a classical outward heat flow). This gives the first formula after standard manipulations. Note that permuting xi with X2 in changes Z2 into —?2 (as it should). Integrating over t gives the second formula. The last result is obtained from the first formula after some tedious but simple computations. One could wonder if this last expression is an honest density. This can be checked as follows. Integrating over t gives, for the first term 4
2 cos(fc7rxi) • - ^
^
=-2xi-f 1
koddyO
by standard results on Fourier series. The second term gives —2x2 + 1- The fourth term leads to , „, 2 v-^ cos(A;;r3;i) ^ ^ 1 ,, , keven>0
Xodd^O ^
'
The last sum above easily gives . ,, >, v ^ sin(A;7ra;2) >
sin(A'7ra:2) ^^ • /i. ^ — = —sm(fc7rx2)
X'odd^Q
by Fourier series theory. We thus have 4 - -
^^ Ya fc euen>0
cos(A;7rxi)sin(fc7rx2) H r ^
^
2 = —
v^
,.,-,,
M
^ smM^i+X2 fceven>0
•\- sin[^7r(a;2 — xi)]]/A;
70
CHAPTER 3
Assume, without loss of generadity, that xi > X2. By Fourier series results, this last reduces to 212• Similarly, the third term of our original expression leads to (—1 + 2a;i). Collecting results, we see that we have indeed an honest density — that integrates to 1. • By the first Theorem above, the original hitting place {Zm) and hitting time (Tm) are given by m
m^
Indeed, Z and T are (almost surely) continuous functionals of W. The weak convergence leads to our asymptotic approximations. Note also that the particular case a;i = a;2 = 0 leads to results of Flajolet[25], Th.2, and our last expression above goes into the bivariate 0 distribution announced in Flajolet, Th.4. R e m a r k . In this section, we have presented an aspect of the colliding stacks problems. Various situations have been analysed by Louchard. Maier[64] considers the case where the probability of insertions into and deletions from a stack may depend on the height of the stack as a fraction of m. His approach involves large deviations techniques. More general results about exhaustion of shared memory are available in Maier-Schott[65]. The probabilistic technique involved in [65] has also been applied in [66] in order to quantify the rate at which regular approximations to shuffle products of context-free languages approach the original languages. 2.3
THE BANKER'S ALGORITHM
In this situation the absorption time T^ corresponds to deadlock detection. This can be modelled by a random walk in a rectangle with a rectangular section removed from the upper right corner. As long as the random walk remains inside the main rectangle the banker satisfies requests without delay, but inside the sectioned-off rectangle prevention is necessary since the customers Pi and P2 violate the announced upper bounds mi and 17122.3.1 The trend-free case As in the storage allocation algorithm, several subcases must be considered. To simplify the analysis, we will set mi — m2 = m. Here consider the isotropic case pi = qi = P2 = q2 = 1/4. The classical reflection principle across the absorbing boundary is applicable and leads to another diffusion problem. The problem is now to find the behaviour of the random walk inside these domains. The density f(x,y,t) :— F j:[W{t) € dy,t < T] can be found using partial differential equations methods. Let A n FQ Fr 9n •^ 6
denote the two dimensional {x,y) Laplacian operator. be a 2d-polygonal domain depending on the problem under study. is the absorbing part of the boundary of H. is the reflecting part of the boundary of fi. = Fa U Fr is the boundary of il. represents the outward normal derivative to 5 0 . is the two dimensional Dirac distribution.
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
71
Then the density satisfies
Of = AF, dt ' f- = 0, df = 0, du ' fix,y,0)
= S{x -
in Q, \/t, along Ta Vi, along Tr xo,y~yo)
To solve this problem, we separate t from the space variables by setting
n=l
where £>„ are real coeiRcients to be chosen in order to satisfy the initial condition. The eigenvalues A„ and the eigenfunctions M„ axe solutions of a classical mixed homogeneous boundary value problem for the 2d-Helmholtz equation: Aun + A„ii„ = 0
in fl
M„ = 0
along Fa
-;:— = 0
along Fr
Of
To find u„ and A„ we can make use of the large singular finite element method. This method has been specially developed to tackle elliptic boundary value problems in polygonal domains, Descloux-Tolley[14]. The particular case of the Helmholtz equation has been studied, both from analytical and numerical points of view, in Descloux-Tolley [14]. As shown by them, the eigenfunctions Un may be written as: oo
" " = '^CinpJp^i^iy/KriJglfXipdi),
for {ri,di) € fii
where f2i is the closure of the «'*' subdomain 0.,, i = 1,2,-• • ,N that constitute the subdivisions of ft, ft = fti U ft2 U • • • U ftiv. Every ft; contains at most one vertex of 5ft and ft, n ftj = 0 for (i ^ j). Moreover, aft,- n 9ft> = r,^ for i ^ j , with Tij = 0 if it is formed of a finite number of points. ri, 9i are polar coordinates centered at a point P; lying on 5ft fl 5fti. In general, P; is located at a vertex of 5ft Cinp i = 1,2,... ,N, p = 1,2,... are real coefficients to be properly chosen. More precisely, the C'inp should be taken in order to ensure the continuity of «„ and of its normal derivative along r,j, i ^ j ; i,j = 1,2,... ,N.
72
CHAPTER 3
Hip are real numbers whose values depend on the particular boundary conditions Jfii^{-) is the Bessel function of the first kind and of order Hip g(-) is a sine or cosine function, depending on the boundary conditions. Finally, taking into account that for A„ / A^ I/
=
Ur,{x,y)u,riix,y)dQ I/ U„(X,yjUm{X,y)dil
= •{ I ||^_ i' 11||2 =
IL
6(x - xo,y - yo)M„(a;,y)rffi =
ii m ^ n -^ ^
u„{xo,yo)
we obtain f(^ „ /^ - V J{x,y,i)-2_^
''n{xo,yo)u„{x,y) II II2
i^^^,
R e m a r k . The separation of ft into subdomains Q,i can be done in sever2d ways. However, it is important to take into account the convergence properties of the series. For some eigenvalue A„, with the proper coefficients Cinp, the series converges in a disk centered at Pi with radius Ri > 0, where Ri is the distance from Pi to the "nearest singularity of u „ " . We shall not discuss here the precise meaning of this. Note that if we define Ri = mindistance {Pi, Pj)
{i,i = 1, 2, • • •, A'')
then Ri < Ri. For practical applications we can thus use Ri instead of Ri. As a consequence, ft should be divided so that every ft, lies entirely in a disk centered at P, with a radius pi < Ri. (ft; contains at most one vertex of ft). For the deadlock problem for the two dimensional banker algorithm, one is working in a five-sided domain consisting of a square with the northeast corner cut off'. This leads to eigenfunctions of the form Y^'^=i C,„pJ2p-2(y/Kri)
cos{2p - 2)0i,
E^lC'3np-^(4p-2)/3(V^'"3)sin^^^3, 2 ^ 1 C'4npJ(4p-2)/3(\/^''4)cOS -^^-9^,
{ri,0i) € ft,, i = 1,2,5 (''Si^s) € fta {ri,9i) € ft4
for a suitable decomposition into subdomains. If we take as initial condition a Dirac 6 at the origin 0 (ZQ = yo = 0), we have _ w„(0,0) _
Ci„i
and fix,y,t)
= y2 II ^"u2
u„(x,y)e~i-
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
2.S.2 Numerical
73
results
In general, the coefficients Ci„p and the eigenvalues A„ cannot be computed by an explicit formula. These numbers can be found by solving a generalized eigenvalue problem for an infinite linear algebraic system (see [14]), which is unwieldy in the present case. In order to obtain an approximate solution, we truncate the infinite summations. The exact solutions are then replaced by the approximations
where Pi
Un = /^ CjnpJ,ii^{y)^nri)g{tJ.ip6i) p=l
for {ri,6i)
e H;
The approximate values Cjnp and A„ of Ci„p and A„ are found as explained in [14]. Numerical investigations show that it is possible, though not easy, to obtain accurate approximations to the eigenvalues A„ and eigenfunctions M„. The main dificulty is due to the presence of the Bessel functions that lead to very ill-conditioned matrices if not suitably scaled (as noticed by Fox, Henrici, and Moler). Another numerical problem arises from the initial condition. In order to have a sharp approximation of the initial condition in the form of the generalized Fourier development TV
f{x, y, 0) = 8{x, J/) ~ ] ^ DnU„{x, y) n=l
it would be necessary to have a very high value of N. This is practically impossible due to the enormous amount of calculations required. However, as the values of A„ (and A„) increase rather quickly with n, even for small values of t, / ( x , y, t) may be considered as a fairly good approximation to / ( x , y,t) due to the rapid decay of e""^^"'. As a criterion to test the validity of approximation one can evaluate
-\fdtl
'Ids
and compare this quantity to the exact value
2 Jo
ir. du
R e m a r k . With a; = 0 the deadlock problem for the banker algorithm degenerates into that of the evolution of two stacks in bounded space whose solution is well known, see Louchard-Schott [61].
74
CHAPTER 3
The results of some numerical computations is reported in the article by Louchard, Schott, Tolley&Zimmermaii[62]. R e m a r k . These methods apply to any kind of polygonal domain and of course to triangles (i.e., the 2 stacks problem). The question remains how to solve the general problem. It looks like the exit-time technique of Matkowsky et al.[70] is applicable and will give at least the asymptotic behavior of (T), the expected value of the hitting time, cf. Maier[64]. The question remains: How to get the limiting distributions of T and Z in the general case? 2.S.S Tree intersection
problem
We would like to mention the paper On the average size of the intersection of binary trees, by R. Baeza-Yates et al.[8]. The interesting feature is that Bessel functions play an essential role in their analysis. R e m a r k . The classic example of the analysis of an algorithm that involves Bessel functions is Jonassen-Knuth[46]. For some probabilistic methods applicable to the sorts of problems discussed here, see Aldous[4].
BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS
75
Colliding stacks
i£I
R
'b
(t-
R
1 <^
R
Banker algorithm
Chapter 4 F O U R I E R T R A N S F O R M ON F I N I T E G R O U P S A N D RELATED T R A N S F O R M S
I.
Basic Definitions and P r o p e r t i e s
A group is a set with an associative operation, an identity for the operation, with the property that every element has an inverse. We will conventionaJly denote the operation as multiplication, and 1 for the identity. Typical examples of abelian, i.e., commutative, groups include rings and fields, with respect to addition. Nonabelian groups that arise naturally are permutation groups and groups of geometric transformations. A group action on a set S means a mapping from G to the set of invertible functions on 5 , which form a group under composition. Denote the mapping by the correspondence g —^ (f>g. We can also consider this as a map G x 5 —> 5 : {g,s) -^ (t>gis) The mapping must satisfy two conditions: a. Homomorphism: (j)ggi(s) = (j)g((j)gi(^s)), i.e., 4>gg' = 4>9° 4'g' h. Identity: 4>-i{s) = s, i.e., the identity of G goes to the identity map on S. Condition a. can also be thought of as an associativity condition. Writing g • s for the action oi g on s, we have (gg') •s=g-ig' -s) If the set 5 is a vector space of dimension d < oo, and G maps to linear transformations on S, g —> Tg^ then the action is a repTCsentation of G of degree d. A simple example is the trivial representation in which every element of G maps to the identity operator on 5 . For a given basis on S, the elements of g are realized as dx d matrices. Representations T^^' and T'^^ such that, for all g e G,
for some invertible operator Q are equivalent. Realized as matrices, these differ only in choice of basis. It is usually understood that referring to representations means up to equivalence.
FOURIER TRANSFORM ON FINITE GROUPS
77
1.1 Definition. If S is a vector space over C with an inner product { , ), then a representation is one such that the operators Tg are unitary for every g:
unitary
(T,v,T,w) = (v,w) for all V, w € 5 . An observation that is often useful is that for a finite group G, any given representation is equivalent to a unitary representation. This may be seen as follows. Let [u, w] be any inner product on 5 . Given the representation T, define
The representation T is unitary with respect to this inner product, since ( T , v , r , w ) = jL ^ [ T , , T , v , r , - T , w ] \G\ ^
which reduces back to (v,w} since gg' for fixed 5 is a permutation of the elements of G. Now, we know from linear algebra that ( v , w ) has the form [ P v , P w ] for some invertible P . Thus, T is equivalent to T defined by Tg = PTgP~^, which is unitary with respect to the original inner product. The character
of the representation T is the function on G given by the trace
x{g)^trTg which depends only on the equivalence class of T. The character of the trivial representation is just dim 5 , the degree of the representation. A fundamental role is played by irreducible representations. A representation is irreducible if there is no non-trivial subspace of S that is invariant under the group action. That is, you have no representation on a nonzero subspace strictly smaller than S. Over C, every representation of a finite group is covnpletely reducible that is, the space S on which it acts can be written as a direct sum of invariant subspaces on each of which G acts irreducibly. The matrices of the representation take a common block diagonal form corresponding to such a reduction.
78 1.1
CHAPTER 4 G R O U P ALGEBRA AND F O U R I E R T R A N S F O R M
The idea of the Fourier transform on groups is to use representations to study functions on the group. Here we will only be concerned with the case of \G\ £nite. One thinks of functions / : G —> C as elements of r ( G ) , the group algebra consisting of formal sums over G with coefficients given by the corresponding functions:
This is an alternative way of writing the |G|-tuple {f{g))g€G, displaying the values of the function / . The principal feature of the algebra F is that multiplication in F corresponds to convolution of functions. 1.1.1 Definition.
For functions / i , /2 on G, the convolution
/ i * /a is given by
gea
1.1.2 P r o p o s i t i o n . elements in F. Then
Let fi,
/a be functions
on G, 7 ( / i ) , 7(/2) ihe
corresponding
7(/i)7(/2)=7(/i*/2)
Proof:
Write
7(/i)7(/2)= J2 f^^9')f2(g")9'g" a' ,9"
and change variables g' = xg ^, g" = g. The result follows.
•
This is the 'generic' version of the Fourier transform. For any representation T, we have corresponding to F the operator algebra generated by { Tg }geG within the algebra of linear transformations on the given vector space. Since the product of any two operators of the form Tg is again of the same form, we have the elements of the operator algebra of the form gee
for some scalars Cg. This is exactly the im.age of 7 ( / ) € F corresponding to the function f{g) — Cj- Thus, for any representation T, we define the Fonrier transform of / by
f(T) = X; f(g)Tg gee
FOURIER TRANSFORM ON FINITE GROUPS
79
Taking traces gives the Fourier transform of / evaluated at the character %
fix) = E /(^)x(s) Suppose that we can parametrize the irreducible representations, call them T^^\ corresponding characters x^- Then we write
with
/(T(^)) = Yl f^9)Ti^^ gea
m = J2f(9)xH9) gea
1.2
CLASS FUNCTIONS AND SCHUR'S LEMMA
There is an important feature of the algebra of characters. Since xis) = t r T j satisfies Xihgh-')
= tr ( T A T . T ^ - I ) = t r T , =
xig)
the characters depend only on elements equivalent up to conjugation, namely on the conjugacy classes. We denote a typical conjugacy class by p. 1.2.1 Definition. We denote by x^ the value of the character of an irreducible representation T^^* on the conjugacy class p.
1.2.2 Definition.
A class function
on G is a function on the conjugacy classes of G,
fig) = for all
fihgh-')
heG.
Equivalently, a class function is determined by the property that figg')
=
fig'9)
for all g, g' € G. It follows readily from the definitions that 1.2.3 P r o p o s i t i o n .
The convolution
of two class functions is a class
function.
Another terminology is central functions . Recall that the center of G consists of those elements commuting with all other elements. Similarly, the center of an algebra consists of elements commuting with all others. Elements of the center are called centrai. 1.2.4 P r o p o s i t i o n . / is a class function if and only if'f(f) of the group algebra F, i.e., 7if)7i) = for all functions on G.
li4>)lif)
— ^ f{g)g is in the center
80
CHAPTER 4 Proof:
This is the same a,a f * (j) = * f ior &\\ :
9€G
gea
Write xg^^ = k in the second sum to get, with g = k~^x
k€G
keG
for / central. Choosing (f>{k) = 1 if fc = g'~^, 0 otherwise, the above equation reads • fig'g) = figg'), yielding the converse. Now we have the fundamental 1.2.5 L e m m a .
Schur's
Lemma
a. Let U commute with all of the operators of an irreducible representation is a scalar multiple of the identity. b. Let T and T' be irreducible representations
T. Then U
such that, for a nonzero U,
UT = T'U Then T and T' are
Proof:
equivalent.
For a., note that any eigenspace of U will be invariant under the Tg-. t / v = A;V => VTgV = TgUV = kTgV
If U is nonzero, then it has at least one nontrivial eigenspace, which by irreducibility must be the whole space. For b., let V be in the kernel of U, i.e., Uv = 0. Then T'Uv = UTv = 0 so that the kernel oi U is an invariant subspace for T', if J7 ^ 0, then it must be zero. Hence U is one-to-one. Similarly, the image of U is invariant under T, so that U is onto. I.e., U is an isomorphism and the representations are equivalent. • Thus, any element of the center of the group algebra, F, maps to a scalar multiple of the identity for any irreducible representation. I.e., 1.2.6 Corollary. For any class function f on G, f{T) is a scalar multiple of the for any irreducible representation T.
identity
For a conjugacy class p, denote the fimction which is 1 on ;0, 0 otherwise, i.e., the indicator function of p, by | p .
FOURIER TRANSFORM ON FINITE GROUPS
1.2.7 Definition.
81
We denote by 7(p) the element of the group algebra lip) =fi^p)
=
^ 9 9Gp
Now, any class function is of the form / = 1.2.8 P r o p o s i t i o n .
Ylpf(p)(p-
For any irreducible representation
T^'^', of degree dx, we have:
UT''^) = \P\fx'pI ipW = \p\x). and for any class function
f, fiTW)
=
l.f^X)I
Proof: By Schur's Lemma, ^p(T'^') is of the form cxI for some scalar ex- Tailing traces gives
aA) = ^ t r T ( ^ ' = Hx^ SGp
= tr {cxI) = cxdx which gives cx = <^^^|plXn- Thus the result for (,p and hence the result for any class function / via J(p) = '£,f{p)(p• Observe that this Proposition is a refinement of the Fourier transform of a class function:
/(A) = Y, \p\fiP)^ 1.3
REGULAR REPRESENTATION AND CHARACTERS
The regular representation corresponds to the representation of G acting by multiplication on the group algebra F. We denote this action by 7(3):
y
y
This action can be expressed directly as an operator on functions:
In general, as noticed above, multiplication corresponds to convolution. For any / we define the corresponding convolution operator on the space of functions on G Tfc^ = f* corresponding to the multiplication by 7 ( / ) in F.
82
CHAPTER 4
1-3.1 Deflnition. given by
The standard inner product on F, equivalently on functions on G, is (7(/i),7(/2)) = (/i,/2) = E
hiaWj)
gea
One readily checks: 1.3.2 P r o p o s i t i o n .
The adjoint of the operator Tf is the operator
T}=Tf, vhere /+(x) = / ( x - i ) We have 1.3.3 Corollary.
The inner product
can be expressed in terms of convolution as
We want to look at this in terms of the action of 7 ( / ) on F. R e m a r k . In the following, by tr we mean the trace of the indicated operator for the regular representation. Note that the standard basis for the space F is given by the elements 7(g). 1.3.4 L e m m a .
For the regular representation tr7(/) = /(l)|G|
Proof:
Consider multiplication by 7 ( / ) on the basis vector "/{g):
X
X
The contribution to the trace is given by the coefficient of g, corresponding to the diagonal element of the matrix of 7 ( / ) on F. This gives / ( I ) . Summing over g € G yields the result.
• 1.3.5 T h e o r e m .
We have the
isometry
(/i,/2) = | G | - l t r ( 7 ( / i ) 7 ( 4 ) )
FOURIER TRANSFORM ON FINITE GROUPS Proof:
83
Combining the above observations: (/i,/2> = (/i * / l ) ( l ) = | G r H r ( 7 ( / i ) 7 ( / j ) )
as required.
•
Now consider the operators Tp, the convolution operators corresponding to multiplication by 7(p) for the conjugacy class p: f{Tp) = jiip * 4,) = 7(p)7(^) We denote the operators corresponding to the ^^ by
Now observe 1.3.6 P r o p o s i t i o n . If JS € F is central, then the eigenspaces of z are invariant under the group, i.e., multiphcation by 7(3), for all g e G.
Proof:
If 2V = Av, then, similarly as we saw in the proof of Schur's Lemma, 27(9')v = 7(fl')2V =
\j{g)v
• Now we have the decomposition into eigenspaces of the 7(p), the characters appearing in the eigenvalues. 1.3.7 T h e o r e m .
We have the orthogonal direct sum decomposition
ofT
A
where each Mx is an eigenspace of the 7(p) with
-yiph =
-^Xpf
for each 7 in an invariant subspace of M\ corresponding rp(\)
to an irreducible
representation
84
CHAPTER 4
Proof: The adjoint of Tp is given by convolution with if J, equivalently by multiplication by ~f{py. Thus, the adjoint of Tp is an element of F. Since the jip) are central, in particular, it commutes with -f(p)^ • Thus, the operators corresponding to multiplication by the 7(p), equivalently, the convolution operators Tp are a comm.uting family of normal operators. Hence they are simultaneously diagonalizable. This gives the indicated decomposition. The eigenspaces are invariant under the group, so contain irreducible representations. By Prop. 1.2.8, for the irreducible representation T'-^\ each 7(p) has the indicated eigenvalue. • R e m a r k . Note that the trivial representation M{i} has basis 7(u) = 2 G 5 ' corresponding to the constant function u{g) = 1 for all g. This element is invariant under multiplication by all g eG. So the M\ contain copies of the irreducible representation T^'^K Denote the projections eA:r —> M A . Then t r e ^ = nixd\ where m\ is the multiplicity of the irreducible representation T^'^' in Mx. It turns out that rtix = dx- We can now show 1.3.8 T h e o r e m . We have the
Fourier
inversion
expansion
/(s) = ^E'^^tr{T'_^>/(T(^))}
on
Proof: Observe that, in general, g acts on Mx as T j ^ ' so that 7 ( / ) acts as /(T^^^) i.e.,
MA,
7(/) = 5]/(T(^>)eA A
Now, 7 ( / ) = E / ( ? ) S , so that /(fir) = (7(/),7(S'))- To apply Theorem 1.3.5, note that ligV = fig-')Thus /(ff) =
^tr(7(/)7(ff)^)
=wi^-^'^{^i-y(T^'')) 1^1 . and the result follows from the equality mx = dx-
•
From this theorem, we can find the form of the projections ex 1.3.9 P r o p o s i t i o n .
Tije projections ex are given by
FOURIER TRANSFORM ON FINITE GROUPS
85
Proof: Thus,
\.
By definition, e\ acts as the identity on Mx and annihilates M^ for fi ^
By the Fourier inversion formula, writing ex = "^{(ftx)
M9)=~-.Y.<^,tr{Tll\Sx,)
By equivalence to a unitary representation, we have that X\9'') = X^) Hence the result.
(1.3.1)
•
Observe that we have, then,
e.= | i 7 ( ^ ) 1.3.1 Schur orthogonality
relations
Using Schur's lemma we can show the orthogonality properties of the matrix elements from which orthogonality relations for the characters follow. 1.3.1.1 T h e o r e m .
Orthogonality of matrix
Denote the matrix elements ofTg by tij{g). tions. Then
elements Let T and T' be irreducible
representa-
a. IfT and T' are inequivaient,
9€G
for all choices of indices i,j, k, I. b. For a given irreducible representation
i.e., they form an orthogonal system. {tki,tij)
T of degree d,
For unitary representations, =
d~^\G\6ij6ki
we write
86
CHAPTER 4
Proof: For each transformation A of appropriate size (if T and T' are of different degrees), define the intertwining operator ^T,AT;..
L{A) = gee This intertwines the representations T and T':
T,i(A) = 5^r,,Ar;_. g€C
geG
Thus, by Schur's Lemma it is zero if T and T' are nonequivalent while otherwise it is a scalar multiple of the identity. Letting A = E^^ be the matrix having a nonzero entry equal to 1 in the ij position, we get the kl^^ entry
see Thus, a. follows. For b., taking T = T' gives
gea
Summing over k = / yields, d denoting the degree of the representation, dc'^ = 6ij\G\ using the representation property and the fact that Tg-i = T~^. The result follows. Now we apply this to find the orthogonality relations for the characters. 1.3.1.2 T h e o r e m .
Schur orthogonality
a. For irreducible representations
relations
\, A' {x\x^')=Sxy\G\
i.e., they are orthogonal unless equal. b. For conjugacy classes p, p', 2_j ^p>^p' = °PP' TTT
FOURIER TRANSFORM ON FINITE GROUPS Proof:
87
To prove part a., the orthogonality of the matrix elements says
seG
Summing over k — i and j = I gives
Using the relation, eq. (1.3.1), x^'ig ^) = X^ig) this reads {x'^,x^') have, using the formulation for unitary representations, {tki.ttj)
=
= 0. For A = A', we
d~^\G\SijSki
Summing over k = i gives on the right hand side d~^\G\6ji and summing over j = I yields the result. For b., apply Theorem 1.3.5, first observing that multipUcation by fip)\ as the adjoint to multiplication by 7(/9), acts on Mx by multiplication by the complex conjugate eigenvalue. Thus,
\G\Wpp' =
\GMp),jip'))
= tr(7(/>)7(p')n
from which the result follows.
•
From part a. of the Schur orthogonality relations we can prove the result mx = dx that has been used in the proof of part b. and has been useful above as well. 1.3.1.3 P r o p o s i t i o n . The muJtipJicity mx of the irreducible representation regular representation equals the degree dx-
Proof:
We have by construction that
A
By the first Schur orthogonality relation, we have
On the other hand, we have already seen that
T*^' in the
88
CHAPTER 4
unless g = 1, in which case it equals \G\. Now, for any representation, x ( l ) is the degree. Thus, see as only one term is nonzero, hence the result.
•
R e m a r k . The matrix with entries Xp is called the character table of the group G. It is customary to label the rows by the irreducible representations and the columns by the conjugacy classes. 1.3.1.4 P r o p o s i t i o n . The number of irreducible representations of conjugacy cJasses. I.e., the character table is square.
Proof:
is equal to the number
From the Schur orthogonality relations we have
|G|E^p>^p
IH
Multiplying both sides by \p\ and summing over conjugacy classes yields, C denoting the number of conjugacy classes,
which equals the number of irreducible representations. 1.3.1.5 Corollary.
•
T i e degrees of the irreducible representations
Proof: We have noted that t r e ^ = fnxd\ regular representation,
= d^.
satisfy
From the decomposition of the
|G| = t r l = ^ t r e A A
as required.
•
In succeeding sections of this chapter we will look at some representations of abelian groups. Then in Chapter 5 we will look at topics related to representations of the symmetric groups.
FOURIER TRANSFORM ON FINITE GROUPS
II.
89
Fourier t r a n s f o r m on finite abelian g r o u p s
For abelian groups, all elements commute so that each element is its own conjugacy class. Thus, there are \G\ irreducible representations. They are given as one-dimensional unitary representations, homomorphisms from G to the unit circle in C. These are called the characters of the group G. (Since these are representations of degree 1, this is consistent with the terminology for nonabelian groups, but it is a more restrictive usage.) The simplest abelian groups are the cyclic groups having a single generator x. A finite cyclic group is determined by x such that x = 1 for some positive integer TV. The elements of such a group can be enumerated as gi = x', for i = 0,1,... ,N — 1. An infinite cyclic group is isomorphic to the integers Z, with elements x", n £ Z. The fundamental theorem on abelian groups states that a finitely generated abelian group is a direct product of cyclic groups. Thus, the characters of a finite abelian group factor into repesentations of cyclic groups
x{g) = XnA.9)xnM---Xnr{g) corresponding to the factorization of the group. Here we will focus on representations of finite cyclic groups and the connection with finite Fourier transform. 2.1
REPRESENTATIONS O F CYCLIC GROUPS
Denote by C the N^^ root of unity
The characters of the cyclic group with generator x satisfying x^ mappings Xk'-
= 1 are given by the
for fc = 0 , 1 , . . . , iV — 1. Thus, Xk{x') = C*'- Observe that the -conjugate character Xk = XN-k2.1.1
Circulants
The group algebra of the cyclic group with generator oi N X N circulants, so called because successive rows of permutation; i.e., each row has the same entries as the If we map fO 1 0 . . . 0 0 1 ... U= 0 0 0 ... \1
0
0
x satisfying x = 1 is the algebra the matrix are generated by cyclic first row but permuted cyclically. 0\ 0 0
0/
CHAPTER 4
90
then corresponding t o the function on the group such that f{x^) = Oj is the matrix ON-l
\
ajv-2
f{U) =
O.N-2
OiV-l
Oo
aN-3
ffli
a2
as
flo
V
/
R e m a r k . Note that this is actually the transpose of the regular representation: /O 0 0 . . . 1 \ 1 0 0 . 0 1 0 . \0
0 0
0/
On the k irreducible representation, x acts as multiplication by { so that a basis for the representation is given by the vector, t denoting transpose,
(i,C\C",...,C<^-^>'=)' which is cyclically permuted by multiplication by C*. The group algebra is diagonalized given by the unitary matrix !F with entries C^/vN, by the finite Fourier transform 0 < 2,j < A ^ - 1
/I 1 1 jr = N-'/' \1 2.1.1.1 P r o p o s i t i o n .
1 C C^
1 C^ C*
1
\
/•N-1
'
^2{N-1)
c /
C'^~^ ^2(Af-l)
We have the relation U =
TAr*
where A is the diagonal matrix with entries 1, C, C^i • • • > C''^"^ • Proof:
The entries of TA.T* are
where the delta function here means that 1 + i = j
(mod TV).
For a function / on the group, we have the Fourier transform
/(fc) = iv-i J]c*'/(C') The Fourier inversion formula here is
Comparing with the above Proposition, we have
FOURIER TRANSFORM ON FINITE GROUPS
2.1.1.2 P r o p o s i t i o n .
91
For any function / on the group,
/(t7) = 5]/(fc)l/* In Chapter 5 we will see the connection with inverses of circulants and symmetric functions. III.
Krawtchouk p o l y n o m i a l s a n d finite probability t h e o r y
We will look at random walks and martingales for finite probability distributions. Orthogonal polynomial systems for the multinomial distribution will be found. A particular class of generalized Krawtchouk polynomials is determined by a random walk generated by roots of unity. Relations with hypergeometric functions and some limit theorems are discussed. This presentation is based on Feinsilver-Schott[19].
3.1
KRAWTCHOUK POLYNOMIALS
Krawtchouk polynomials Ka{x, N) are polynomials orthogonal with respect to a binomial distribution. For convenience we consider Bernoulli random variables taking values in ± 1 with equal probability. They may be defined by the generating function N
Giv) = J2 ^"K„ix,
iV) = (1 + v)^'^+">/^(l - z ; p - - ) / 2
0=0
The generating function oo
n
/
(l_,).-(l_(l_iJ)„)-v = ^^(„)„^i.J-'^'!/ n=0
with (a)„ = r ( a + n)/T{a),
•
^
\
ji\
(3.1.1) '
yields the identification
A theory for multivariate 'Bernoulli-type' polynomials is presented in [17]. Here we take a different look at generalized Krawtchouk polynomials . One class arises from a random walk on the lattice N [ l , ^, ( ^ , . . . , C''"^] where C — g2jri/d There are evident connections with the finite Fourier transform. It is quite likely that the classes of polynomials discussed here will prove useful in image processing, among other possible applications.
92
CHAPTER 4
In the next section we give the probabihstic approach to the binomial case. Then we present a general approach for finite probability distributions. After that, some related limit theorems are presented. Specialization to the cyclic case completes the study. R e m a r k . Note the following: 1. Throughout, N will denote 'time' — the number of steps taken in the random walk. 2. We use () to denote expected value, e.g., (X). 3. For brevity, we call Krawtchouk polynomials simply K-polynomials . 3.2
BINOMIAL CASE
Consider a random walk on Z, with equiprobable increments ± 1 . We write Xj, I < j < N, for the corresponding Bernoulli variables. The generating function
G{v) = n ( l + vX,) = (1 + ^)(^+^)/2(l - t;)(^-^)/2 where x — Y^Xj is the position after N steps. As noted above, G(v) = with Krawtchouk polynomials Ka-
Y2v"Ka(x,N),
One can take the viewpoint of 'quantum probability' and consider the Xj as the spectrum of an operator A, an TV x iV matrix. Then the condition on A is that A e spectrum (A) => A € { —1,1 }. And the variables of interest axe the multiplicities. We have G{v) = det(l + vA) And the variables x, N are simply ti A, ti A'^. We thus have the principal observation that since the Xj take two values, two variables suffice to specify the Ka, which are seen to be elementary symmetric functions in the Xj. The variables x, N are the corresponding power sums: a; = ^ X j , iV = ^ Xj . The generating function G{v) as a function of A'^ is readily seen to be a martingale, as the Xj are independent with mean zero. It is, in fact, the prototype exponential martingale. The calculation {G(v) Giw)) = J J ( 1 + {v + w)Xj + vwX])
= (1 +
vwf
exhibits the orthogonality of the Ka nicely. So the Ka are important mainly for these two features: 1. They are the iterated integrals (sums) of the Bernoulli process. 2. They are orthogonal polynomials with respect to the binomial distribution.
FOURIER TRANSFORM O N FINITE GROUPS
3.3
93
FINITE PROBABILITY DISTRIBUTIONS
The probabilistic approach may be carried out for general finite probability spaces. Each increment X takes d possible values {ifoi • • • i $s } with P{X — ij) = Pj, Q < j < ^, where throughout we will use the convention 8 — d—\. Denote the mean and variance by n and (7^ as usual. Take A" independent copies of X: Xj, 1 < j < N. Define the martingale N
Giv) = ]l{l + viXj-^)) We may switch to the multiplicities as variables. So set N
the number of times the value ^j is taken. We have 6
N
3=0
a=0
this last defining our generalized K-polynomials . One quickly gets 3.3.1 P r o p o s i t i o n . Denoting the multi-index n = ( n o , . . . ,ns) and by Cj the basis on Z^, K-polynomials satisfy the recurrence Ka(n + ej) = Kain)
+ {^j -
standard
/i)Ka-iin)
We also find by binomial expansion 3.3.2 P r o p o s i t i o n .
\k\=a
where |k| = fco + • • • + ks. An interesting connection with the multivariate theory comes in when we consider equations the hypergeometric form. The Lauricella polynomials FB are defined by
-al
-^
(-r)k(b)k w
K5.
•OI'-IK'" ''
with, e.g., r = ( r j , . . . , rj), (r)k = (ri)*:,(r2)i;j • • • ( r i ) t j for multi-index k, also s^ = Sj' • • • 5**, a n d k ! = fci! • • •fc«!.Note that < is a single variable. The generating function of interest here is
a multivariate version of eq. (3.1.1).
94
CHAPTER 4
3.3.3 P r o p o s i t i o n .
If ^o = 0, then
K(r,^-(m V UipjCiY' p / - r , - n F B ( ^ i^„(n)-(-7V)„ 2^
1
n
^'•••'WJ
|r|=a
Proof:
Let Vj = vpj(,j, bj = —rij, t = —N, Sj = Pj ^ in eq. (3.3.1), for 1 < j < <5.
Note that J2""j — '"('••> Y^^j " ^ ~ ^ " ("i + ' •' + "«) = no-
•
As for the binomial case, we may use the power sum variables, e.g. for centered increments, N
6
}=1
j=0
to express the functions Ka- (This will be useful in the cycHc case.) Orthogonality follows similar to the binomial case. 3.3.4 P r o p o s i t i o n . The K-polynomials Ka(nQ,..., the induced multinomial distribution. In fact,
ng) are orthogonal with respect to
{K^K^)=6^^a'"f^^
Proof:
= ( X ^ (Pj + (^ + '^)P}{i3 - A*) + VWPj{ij Thus, {G(v) G{w)) = {l-'rvwa'^)^. as well. • 3.4
p-fYj
This shows orthogonality and yields the squared norms
LIMIT THEOREMS
Limit theorems for products ]7(1 + vXj) are known for Xj the discrete increments of a process. The limits yield iterated integrals of the process. Here we look at limit theorems based on the power sum functions. First, define the normalized values fij = (^j — /i)/(T. Then the central limit theorem says that
*
7=1
^
j=0
FOURIER TRANSFORM ON FINrrE GROUPS
95
converges in distribution to a standard Gaussian as iV —> oo. We look at the power sums of the normalized variables:
B^)'=«-"•£»/"; 3.4.1 T h e o r e m . Let Yj satisfy n, = pjN + YJVN. converges to a standard Gaussian.
Proof:
We have, recalling that ^PjPj
Then, as N -^ oo, Yl^-^o yj0j
— 0,
^ E^^ft = -^ E(Pi + YiVN)Pi = E YiP^ and the result follows from the central limit theorem. 3.4.2 T h e o r e m .
The martingale for the normalized
• variables
N
^<"'=nC+»^) converges to the Brownian martingale
at time 1,
G{v)
vY-v^/2
e where Y is the standard Gaussian denoting the limit of the normalized
Proof:
sums.
Write
=
Y[exp(n^\ogil+vpJ/^/N))
= exp('f;(-l)-i^5.^ where S; are the scaled power sums N~'''^ S'^j/^j- The above Theorem deals with i = \. For z = 2 we have, with rij = pjN + Yjy/N,
N-'Y.^,P]^Y.P^P]+N-^I^Y.YiP]=.l
+ o(l),
since the /?'s are scaled to variance 1. For i > 3, we have
as A'' —> oo. Denoting the limit of Yl'^jl^j
^V ^ ^^^ result follows.
•
We conclude that the corresponding K-polynomials converge to Hermite polynomials in the variable Y.
96 3.5
CHAPTER 4 CYCLIC K-POLYNOMIALS
Now consider a random walk in C , with increments X taking values in the dth roots of unity: 1, (,, C,^,..., C*, with C = g^'^"''', and (^ = d — 1 as above. For simplicity we discuss the isotropic case, all values occurring with equal probability \jd. The distribution of x = position after N steps is an interesting problem in number theory. Namely, to count how many 'paths' lead to the same position x. Here we make some remarks concerning this problem. Again, let rxj denote the multiplicities of occurrences of the C,^. We have the power sum variables: So = " 0 + " 1 + • • • = N
xi = no-\- niC + . -• = 2 j " j C ^ ~ ^ X2 = no + riiC'^ + ... = ^
Xk
rijC^^
no + ni(''+ ... = y^"iCkj
Notice that the Xk are the finite Fourier transform of the variables ( n o , . . . ,ni). we can conveniently go back and forth between the two sets of variables.
So here
From algebraic number theory, see Ribenboim[77, p.266ff.], it is known that for rf = a prime power, 4>{d) denoting Euler's function, the powers 1, C,,(^,..., ^^('')~i form a basis for the Z-module spanned by the ('•', 0 < j < 6. I.e. each sum of the form ^nj(^ has a unique expression as a sum involving (^ for j < (i>{d). The problem is that we have to count how many general sums, involving all the i^-', reduce to the same canonical form, involving only C''T J < i
Proof:
Let d be prime.
Given xo = N and xi = J^WjC"'; the nj are
In Xi, substitute (^ = —1 — ( — ••• — (^ ^ yielding Xi = (no - ns) + C(ni -nsj-i
h C * " \ n 6 _ i - ns)
By the result quoted above, the numbers nj—n^, 0 < j < 6 are uniquely determined. Thus, hn^-i —Sns is known, thence dn^ =no + --- + ns — i/ — N — v their sum, call it f = noH is determined and hence the n j , 0 < j < <5 as well. •
FOURIER TRANSFORM ON FINITE GROUPS
97
Now we look at the K-polynomials for the cycHc case. The generating function is i
N
Giv) = ]l{l+vCr'
=5]t;°A'„
They satisfy the recurrence, cf. Proposition 3.3.1, A'„(n + ej) = Ka{n) + (•'iiTa-iCn) The binomial expansion gives
K.(no,...,n,)=
E n ( l ; ) c ^
The FB representation is found as follows. In the generating function (3.3.1), let Vj — vC,\ I < j <S. With bj = -Hj, I <j < 6, t = -N, we have 6
6
\]vj
= —V
and
y j bj = no — N
With Sj = 1 — i^-', writing ( = (~^ as usual, 1_ ^
„. + sjvj = 1 + v + (1 - C)vC
= ! + «;(•*
for 1 < j << d. Thus, as in Proposition 3.3.3,
r=a
Finally we have the limit result 3.5.2 T h e o r e m . In G{v), scale v —* vn~^l^, XQ —> XQU and, for 1 < k < 6, Xk x/tn*'*'**'''', where {d,k) = greatest common divisor of d and k. Then
G{v) -^ exp I J2 \kld as n —» oo.
{-!)'-'v'x.fk /
98
CHAPTER 4
Proof:
Fourier inversion says that rij = d^^ ^XkC''-
G{v) = J](l + vC)^^''''^''
I-c,
= Y[{1 + «C^)^""'^' j.k
For fixed k, let (d, k) = g. Then, for 0 < j < d, we can write j = Id/g + r, with r < d/g. Thus, jk = l{dk/g) + rk = rk (mod d) So (''/^'-i'-^ j=0
r=0
^r'-./rf
/=0
id/g)-l {d/g)-l
9-1
] ] [ ( ! + K ' ' ' ' ' ' ) = 1 + y9C^''/''>(»(*-i)/2) = 1 _|. y 9 ( _ l ) » - i (=0
With the scahngs indicated, z* —> x^n^' converge to expressions of the form
, v —> vn~^''^,
each factor of the product will
Taking the product over 0 < r < d/g, we have (d/g)-l
r=0
if 3 ^ A;, and = d/g for g = k, i.e. when A: | d.
^tj/
C»-* - 1 ~ •
R e m a r k . It looks like an analysis for general abelian, or partially abelian, groups based on similar ideas should be feasible. It would be interesting to construct K-polynomials associated with graphs and other discrete structures.
FOURIER TRANSFORM ON FINITE GROUPS IV.
99
Appell states
This provides a unified picture of classes of orthogonal polynomials connected with integral transforms. As we saw in Chapter 2, the various forms of moment generating functions correspond to certain integral transforms; the Fourier-Laplace transform and the Stieltjes transform. Here we look at a general approach and then look in more detail at the Fourier and Stieltjes cases. The Fourier-Laplace transform leads to the Meixner classes which are the subject of the next section. 4.1
DEFINITION OF GENERAL APPELL STATES
Given a probability measure, p(dx), we taice a family of square-integrable functions F(s, x). As usual, we denote integration with respect to p as the expected value, and we set, for the given family F{s, •)
Mis) = {F{s,X)) 4.1.1 Definition. are the functions
The Appell states
with respect to the measure p and the family F
*^(x) =
F(s,x)/M(s)
That is, the ^g are the functions F normalized to have unit expectation. The term states comes from physics denoting a function of unit norm in L^ of p. The idea is that the state is a line or ray in the vector space, an equivalence class of functions up to multiplication by scalars. Note here that the ^^ remain invariant if the functions F are multiplied by scalars. Typical choices of the family F are F{s, x) = e"^ , corresponding to Fourier-Laplace transforms, and F{s,x) = (1 — sx)"^, corresponding to Stieltjes transforms. More precisely, the Stieltjes transforms arise from the family l/{s — x), but we use the alternative formulation, as we did in Chapter 2. If X is an element in a Lie algebra and ipo is an element in a representation space of the algebra, the states corresponding to e'^ •tj>Q are called generalized coherent states. The Appell states are used to define transforms of operators acting on functions of x, typically L^ of p. 4.1.2 Definition. by
The Appell transfoTm {Q)ab =
of an operator Q is the function of a, 6 given
(*a,«'6
Thus, these are the normalized matrix elements of the operator Q with respect to the Appell states. The coherent state representation used in Chapter 2 is the Appell transform for the family e'^ . Here is a useful lemma connecting the Appell transform and matrix elements with the family F itself.
100
CHAPTER 4
4.1.3 L e m m a .
For an operator Q, whenever the matrix elements are deSned we have {^a,Q^,) (*„,**)
Proof: cancel out. 4.2
_
(F(a,X),QF{b,X)) {F{a,X),Fib,X))
Write F{s, X) = M(s) ' l ' a ( ^ ) on the right-hand side. The factors •
M{a)M{b)
APPELL STATES AND ORTHOGONAL POLYNOMIALS
If we have a measure p with a finite moment generating function, i.e., Fourier-Laplace transform, we can form the family of orthogonal polynomials with respect to p. And we can consider the expansion of the exponentials e'^ in terms of the system of orthogonal polynomials. The idea is that operators of interest are typically defined in terms of their action on the polynomial basis, whereas via the exponential functions, Fourier-Laplace transforms, one can transfer to functions expressed in terms of the usual polynomial basis x". This is the basis for the study in Chapter 2. Here we present the basic approach for general Appell states. The main feature is that the family F{s, x) has the property of being the eigenfunctions of an operator Xg acting on functions of x: X,F(s,x)
=xF(s,x)
Denote the family of orthogonal polynomials with respect to p by { <^n } with squared norms 7„ = ||<^TI|| . We define the transforms {<j>n,
(4.2.1)
Thus, we have the expansion (in general, under assumption of completeness of the (f)n)
*s = X ] V„(s)?i„(x)/7„ n=0
In terms of the family F, we have oo F{S,X)
= Mis)
J2 n=0
Vn{s)M^)/7n
The orthogonal polynomials satisfy a three-term recurrence of the form ^(pn = C„(t>n+\ + a„^„
+ 6„i^rt-l
with initial conditions _i = 0, o = 1. Observe that the recurrence relation implies (i>i{x) = ( i — ao)/c(,. We have
FOURIER TRANSFORM ON FINITE GROUPS 4.2.1 T h e o r e m .
Let F{s,x)
101
satisfy F{0,x)
= 1, X^F{s,x)
= xF{3,x).
M ( s ) - » X , ( M ( s ) y „ ( s ) ) = CnVn+l + anV„ +
Then
b^Vn-l
with the initial conditions VQ = 1, Vi = CQ^{M'~^ XgM — ao).
Proof: With (J>Q = 1, setting n — 0 in eq. (4.2.1) yields Vb = 1- For general n, write eq. (4.2.1) in the form {<j^„,F{s,X))=M{s)V„{s) and apply Xs to get {„,X Fis,X))
=
Xs{Mis)V„is))
Dividing out M(s) yields {<j>n,X^s) =
Mis)-'X,{Mis)V„is))
Now use the recurrence formula for <j>n on the left-hand side and apply eq. (4.2.1) to get the result. For n = 0, this procedure yields {X,)=M{s)-'X,M{s) But eq. (4.2.1) for n = 1 says {^i,*^) = Vi. Writing X = co^i + an and taking inner products with ^, thus gives M{sy^X,M{s) and the result follows.
= coVi{s) + ao
(4.2.2)
•
We now apply this theory to some important cases. 4-2.1 Stieltjes transforms
and general
systems
For general systems of orthogonal polynomials, we use the states corresponding to the family F{s,x) =: (1 — sx)"^. The theory applies whenever this is in L^ of the underlying measure. For example, extending to complex values, if s = ia, a real, then |(1 — sa;)""^ | < 1 for all real x. Here, the operator X, = A , , which acts as
And As^'(s, x) = xF{s.,x) Theorem 4.2.1 yields
holds as required. We use the notations of the discussion above.
102
CHAPTER4
4.2.1.1 T h e o r e m . Let (1 — sx)~^ be square-integrabie with respect to the probabiJity measure p{dx). Let M{s) = ((1 — sX)~^). Then we have the expansion in orthogonal polynomials 1
•^
= M(S) V Vn(s)M^)hn n=0 ^—i 1 — sx where the coefRcients Vn, n>> 1, the recurrence formula 1, satisfy s. Vnis)
= C„V„+i + anVn +
bnVn-l
-1 fM(s)-l
\
with Vois) = 1 and
Proof: This follows directly from Theorem 4.2.1. We have to check that for n > 0, V„(0) = 0 so that M-'^X,MVnis) reduces to y „ ( s ) / s . We have F(0,a;) = 1 for all x, and M(0) = 1, so that Vl/o = 1. Hence Vn{0) = {n,-^o)=0 follows by orthogonality.
•
An interesting application of this theorem is to the Chebyshev polynomials. As we saw in Ch. 2, Prop. 1.5.3.1, the coefficients of the expansion of (1 — sx)~^ are of the form Vn{s) = V{s)". It is not difficult to see from the above Theorem, that this form of Vn(s) implies a quadratic equation for V{s), which leads back to the form of V(s) we had in Chapter 2 with extra parameters coming from the recurrence relations. 4.2.2 Fourier-Laplace
transforms
and Meixner
systems
Taking the family F(3, x) = e^^ , we have M{s) = (e'''^ ) is the usual moment generating function or Fourier-Laplace transform. The operator Xa here is d/ds, the first derivative operator. We have 4.2.2.1 T h e o r e m . xioxniais
The exponential function e'^ has the expansion in orthogonal poiyoo e^^ = Mis)
where the coefficients V„, n>l,
Y,
Vnis)n(^)hn
satisfy the recurrence
formula
K + CoViVn = C„V„+i + ( a „ - ao)Vn +
with Vo('S) = 1 stnd
hnVn-l
FOURIER TRANSFORM ON FINITE GROUPS
Proof:
103
In the proof of Theorem 4.2.1, eq. (4.2.2), yields, with X, — d/ds, M'{3)/M{s)=coVi{s)
+ ao
This gives the formula for Vi. As well, for the left-hand side of the recurrence, we have = {M'i3)/M{s))V„is)
Misr'X,{M{s)V„i3))
+ KC^)
and substituting back in the expression in terms of Vj, the result follows. Meixner systems
•
arise when we have the special form
Vnis) = v{sr where, in particular, 1^1(5) = V{s). From the above theorem we have 4.2.2.2 T h e o r e m .
For Meixner systems we have the
expansion
00
e- =
Mis)^VisrM^)hn n=0
where
satisfies t i e Riccati difFerential
equation V = f + 2aV + fiV^
and the recurrence formula for the orthogonal polynomials
is of the form
x(i>n = (co + /3n)n+i + (ao + 2an)n + jn4>n-i
Proof:
Substituting V„ = V" into the recurrence given by the theorem above yields nV"-W
+ coV"+^ = c„y"+^ + (a„ - ao)V" + 6 „ y " - i
Dividing out V"~^, rewrite this in the form y, _ c„ — Co n
2 I "n ~ " o y n
, ^ n
Since these coefficients are independent of n, we set Cn = Co + / 3 , and the result follows.
a„ =2na
+ ao,
b„ = nj
•
We will look in more detail at these systems in the next section.
104
V.
CHAPTER 4
O r t h o g o n a l p o l y n o m i a l e x p a n s i o n s via Fourier t r a n s f o r m
The Meixner polynomials are special families of orthogonal polynomials closely related to operator calculus and Lie algebras. A particular class, the Krawtchouk polynomials, can be thought of as functions on the finite abelian group ZJ and thus the calculation of the Krawtchouk transform is of interest. E.g., Diaconis-Rockmore[15] mention the question of rapid calculation of the Krawtchouk transform. Here, we use the close relationship between Meixner polynomials and operator calculus to give expressions for the generalized Fourier coefficients of a function expanded in a series of orthogonal polynomials of Meixner type. From these formulas, we obtain, in conjunction with the fast Fourier transform ( F F T ) , efficient methods for the calculation of the coefficients. First we present the basic facts concerning Meixner polynomieJs and their connection with operator calculus. Next, we give a detailed discussion of the role of the lowering operator V. The Krawtchouk case is discussed in detail. After that comes the general Meixner case, and a return to the Krawtchouk expansions. We reduce the transforms to multiplication by a Vandermonde matrix and the F F T . R e m a r k . See Al-Salam[5] for various characterizations of classes of orthogonal polynomials. Applications of Krawtchouk and Meixner polynomials to the numerical evaluation of integrals involving Bessel functions have been given by Gabutti-Minetti[36]. The discussion here is that of [18]. Also see [16] for the connections between the operator formulation and stochastic processes. 5.1
MEIXNER POLYNOMIALS AND O P E R A T O R CALCULUS
Our operator calculus corresponds to representations of the Heisenberg algebra on a vector space. The realizations of the basic operators for the Meixner polynomials classes may be given explicitly. A set of three operators A, B, C is a. basis for the Heisenberg algebra if they satisfy [A, B] = C and [^4, C] = [B, C] = 0, i.e., C commutes with A and B. Given a vector space with basis «/)„ a representation of the Heisenberg algebra may be given in terms of the raising and lowering operators R and V defined by their action on the basis vectors: R'^n = 1pn+l Vlpn =
ni)n-\
It is readily seen that, with the Lie bracket given by the commutator VR — RV, [V,R]=I I denoting the identity operator.
FOURIER TRANSFORM ON HNITE GROUPS
5.1.1 P r o p o s i t i o n . aigebra.
105
The operators V and R defined in eq. (5.1.1) generate a Heisenberg
The Meixner polynomials are orthogonal polynomials such that V is expressed by an analytic function oi D = d/dx, where x is the variable in which the polynomials are given. After suitable normalizations, one finds six families of orthogonal polynomials as follows with the corresponding functions V (the functions H{s) listed here will be explained in the next section. See volume 1 [21], for complete details.) 5.1.2 P r o p o s i t i o n . form:
For the Meixner classes of polynomials
Meixner Meixner — PoUaczek Krawtchouk Charlier Laguerre Hermite
V{s) -
\— q — a tanh qs
H(s) = - - s - log ^ j3 sinh qs
V(s)=tans
H{s) =
V{s)=tasihs
\ogsecs
/f (5) = log cosh 5
V{s) = e' - I Vis) = s/(l
the V operators take the
H{s) = e' - 1 - 5
— s)
H{s) = — log(l — s) — s
V(s) = s
H{s) = s^ jl
•where, for the general case, a, j3 are given parameters
and q'^ — a^ — /?.
(Note the normahzations V{0) = H'(0) = 0, V ( 0 ) = 1.) 5.1.1 Generating
functions
The role of V as lowering operator may be seen from the generating functions. These have a particular structure that is determined by the function H{s), the logarithm of the moment generating function of the measure of orthogonality. Each class of polynomials corresponds to a convolution family of measures, p(. They satisfy the relation gtH(.s) =
f e'^^ptidx)
(5.1.1.1)
where H is analytic in a neighborhood of the origin (in the complex plane). Let U be the functional inverse of V (in a neighborhood of 0) and set M(.s) = H{U{s)). The generating functions have the form: ^rU(s)-tM(,)
= ^
1 - J„(x,t)
(5.1.1.2)
the function V is given as the derivative of H. (This is discussed in detail below). The measures pt are a convolution family, pi corresponding to the t th power of pi: pt = p**.
106 5.2
CHAPTER 4 THE OPERATOR V
The operator V is the lowering operator, the action of which on the polynomials is given by that is, V acts as a generalized derivative operator. We use J„ to denote J„(x,<). From orthogonality, one finds that V satisfies a Riccati equation, cf. the previous section on Appell states, which in standard form may be written V = 1 + 2aV + PV^ for real constants a, /9, the prime denoting differentiation. Consider the measure of orthogoneJity p{dx) (i.e., pi{dx)). With a replacing is in (5.1.1.1) we express the moment generating function in the form gH(.) ^ I
e^'pidx)
Differentiating with respect to s on both sides we have V{3)e"'''^
= f
e'^xp{dx)
with V{s) = H'(s). By the Riccati equation it follows that repeated differentiation leads to a relation of the form V{s)"e"^'^
= f e''nix) p{dx)
(5.2.1)
Thus, from the Fourier point of view, V" corresponds to the operator of multiplication by 4>n(x)- We will see that i^„(x) is proportional to J„(x, 1). 5.2.1 P r o p o s i t i o n .
The polynomials
defined via (5.2.1)
satisfy
n\ n{x) =
In
Jn{xA)
where -fn = (J^) = n ! ^ " ( l / ^ ) „ .
Proof: Replacing s by is in (5.2.1), Fourier inversion yields the distributional relation, writing p{dx) as p{x) dx, K{x)p{x)
= j e - " V ( i 5 ) " e ^ ( " ) dsl2i, =
V*(Drpix)
^
FOURIER TRANSFORM ON FINITE GROUPS
107
where V''{z) = V{—z), the * denoting adjoint. In fact,
/•
JU.X,l)V'{D)"pix)dx
=
{V"Jm)=n\6nm
Comparing with the orthogonality relations of the J ' s shows the Rodrigues-type formula
v%Drp{x) = -ux,i)p{x) with 7n = ( / „ ) . Now compare with (5.2.2).
•
In the generating function oo
„
^-—' n! n=0
replacing s -^ V{s) gives
In this form the action of V is clear. Multiplication by V{D) on the left multiplies by 1^(5) on the right, resulting in J„ —> nJn-i5.3
KRAWTCHOUK POLYNOMIALS
First we see how the Krawtchouk polynomials fit into the above scheme. Then calculation of the Krawtchouk expansions is considered. For the binomial distribution, the measurept is given by the discrete weights PN(X) = 2~'^(^) with ir = {N + x)/2. We have the moment generating function:
/
e'" PN{dx) = 2 - ^ ( 6 " + e-* ) ^ = cosh'^ s
where the time-parameter t is replaced by the discrete index A''. Thus H{s) = log cosh 6 H'{s) = V(s) =
tanks
If we think of the Krawtchouk polynomials as elementary symmetric functions (with a factor of n\) in the quantities TT + I's and v — l'.s thought of as the steps of a random walker on the line starting from 0, then we have the generating function (1 + t;)<^+^)/^(l - „ ) ( ^ - - ) / 2 = y
^
K„{x, N)
n=0
where N = TT + 1/ is the total number of steps and x — n — i/ is the position of the walker after TV steps. This may be rewritten in the form
'
Now substitute v = t a n h s to get
n=0
108
CHAPTER 4
5.3.1 P r o p o s i t i o n . e " = cosh'^^ y
5.S.1 Calculation of Krawtchouk
=
^
Kn{x, N)
(5.3.1)
expansions
There are two ways of formulating the basic approach: 1) Replace 5 —» is in eq. (5.3.1). This gives
,'^^^^nJ2'os^~Y^"'K„(x,N)
(5.3.1.1)
n=0
where x runs from —A'^,... ,N in steps of two. We have 27r(iV - 2k) 1+N '
0
If / has Krawtchouk-expansion ^ / n - R ' n / " ' j then, denoting the Gnite Fourier transform of / by / , / ( , ) = (Ar + i ) - i ^ e - - " / ( ^ ) X
where the sum on x runs from —N,...,
TV in steps of two, (5.3.1.1) yields
5.3.1.1 P r o p o s i t i o n . / „ = i" ^ f{s) cos^-"s
sin"s
(5.3.1.2)
2) Denote here ^ ds by D. Thus, f{x) = e ^ ^ / ( s ) | o , which may be denoted just e^° /(O), as we will do below. From (5.3.1) we have
e
^Drr , ^ c o s h ^ - " D s i n h » D /(5) = 2 ^ -, f(s) Kn{x,
N)
0
Setting 5 = 0 gives the desired expansion 5.3.1.2 P r o p o s i t i o n .
fix)
= 2 - ^ J2 ^'"^"".'^^ i^° + e - ' ' )'^-"(e^ - e-^ )"/(0) n! 0
As alternative forms we have:
(5.3.1.3)
FOURIER TRANSFORM ON FINITE GROUPS
109
5.3.1.3 P r o p o s i t i o n . For the Krawtchouk sions for the expansion coefficients:
polynomials
we have the following
expres-
i ) As in (5.3.1.2): U = i"Y^cos^stan"s
/(s)
2) The formula (5.3.1.3), written in the form
Observe that the exponent in the first factor may be computed as |(iV — \N — 2n\). 5.4
CALCULATION OF FOURIER COEFFICIENTS
In this section we present two methods for calculating the expansion of a function for the general Meixner case. The first method is adapted for numerical computation, the second for symbolic calculation. R e m a r k . Throughout the discussion Fourier transform is conventionally defined as
where the integral sign indicates integration from — oo to oo. 5.4-1 Fourier transform
method
Let / have the expansion
Then, since V J„ = n J „ _ i and the J„ are orthogonal with JQ = 1 /n = { V " / ) / n ! 5.4.1.1 P r o p o s i t i o n .
We have the
(y"/)=
expression
fe*"^"^V(is)"f(s)ds
110
CHAPTER 4
Proof:
Recall the (inverse) Fourier transform of the measure pt given by (5.1.1.1),
je">^pt{dx) Thus, Using ParsevaJ's formula, for any function g we have, in the sense of distributions,
{g) = f g{x)pt{dx) = 2n j g(s)Mr)ds From the Fourier inversion formula we have the action of the operator
f{x) =
(5.4.1.1) V{d/dx)
Je''^fis)ds
Combining this with formula (5.4.1.1) for calculating expected values yields the result.
• 5.4-2 Operator calculus method From the generating function, (5.1.1.2) we have:
e—'«(^'=f:^j„(^,o n=0
We extend (5.3.1) for the Krawtchouk case to the general Meixner case. 5.4.2.1 P r o p o s i t i o n .
For the Meixner polynomials
we have the expansion
e - = e ' « W ^—' V ^ Jn!„ ( x , * ) n=Q
Using the operational formula e"^ f(x) = f(x + a), put x —* a, s —* D, then x —> 0, a —* x, which gives the expansion
n=0
Thus
FOURIER TRANSFORM ON FINITE GROUPS
5.4.2.2 P r o p o s i t i o n .
For the Meixner polynomials
111
the expansion coefficients are given
by
R e m a r k . The Vandermonde matrix multiplication here is quite easy to implement. The efficiency of the computations may be improved due to the simplicity of the structures involved in this approach. Practical computation of the Fourier transform may be done by standard F F T methods. An alternative approach is to represent the function by a polynomials and use a symbolic manipulation package to implement the operator calculus. Difference methods may be applied directly as well. Algorithms based on Propositions 5.3.1.1, 5.3.1.3, and equation (5.3.1.3) have been implemented by D. Sheeran using the software FOURIER developed by D. Kammler of SIU-C. Diaconis-Rockmore's [15] algorithm for computing the Fourier transform on finite groups depends on having for any irreducible representation in the form of matrices in GL(V), say, a basis for V such that when the representation is restricted to a subgroup, the matrices become block diagonal. Thus, one looks for a chain of subgroups to blockdiagonalize the representations. This approach is in the spirit of the F F T , but formulated so as to apply to nonabelian groups.
Chapter 5 Y O U N G TABLEAUX A N D C O M B I N A T O R I A L E N U M E R A T I O N IN PARALLEL P R O C E S S I N G
I.
R e p r e s e n t a t i o n s of t h e s y m m e t r i c g r o u p
Given n > 0, S„, the symmetric group is the group of permutations of n symbols. Equivalently, the group of one-to-one mappings on a set of cardinahty n. The composition of mappings provides the group law. Here we will discuss the basic features of the symmetric groups and their representation theory. Some applications including MacMahon's Master Theorem and Molien's Theorem will be presented. 1.1
CYCLE DECOMPOSITION
Throughout the discussion, we fix n. We will typically denote elements of S„ as products of cycles. Each element has a unique representation as a product of disjoint cycles. For example, (154) denotes the 3-cycle, i.e., the mapping which sends 1 —> 5, 5 —> 4, and 4 —» 1, and fixes everything else. Observe that the inverse of the n-cycle {aia2 . • • a„) is (a^an-i . . . a i ) . In particular, any 2-cycle, or transposition, is self-inverse, an involution. To multiply cycles, we interpret them as mappings and multiply by composition from right to left. Thus (1234)(1325) = (14)(25),
(12)(23) = (123),
(145)(25) = (1452)
Notice that fixed elements, i.e., 1-cycles, are typically not written explicitly. If one cycle acts on another by conjugation, the result is that of applying the substitution to the cycle acted upon. An example illustrates this: (532146)-i(524)(632146) = (641235)(524)(532146) =(316) 1.1.1 P r o p o s i t i o n . The conjugacy classes are determined according to the sitions into disjoint cycles.
decompo-
We denote by (n""
...2"'!'")
the conjugacy class of elements having pj j-cycles. Observe that YliPj
— '^•
If we decompose into 2-cycles, then we do not have, in general, a unique decomposition, but the parity of the number of transpositions occurring is well-defined. This only depends on the conjugacy class, since, as is easily checked, an n-cycle has parity (—1)"""^, and the parity is multiplicative. We thus define the class function sgn(a;) = sgn(p) to be the parity of the element x in conjugacy class p. As this m a p is multiplicative, it defines a representation of S„.
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
1.1.2 Definition.
The alternating representation
113
of Sn is the mapping x —> sgn (x).
Acting on n symbols we have 1.1.3 Definition. The antisymmetric representation of S„ means that each element acts by permutation times its parity. The symmetric representation is given by the usual action as permutations.
1.2
YOUNG TABLEAUX AND IRREDUCIBLE REPRESENTATIONS
A Young frame or shape is given by an n-tuple of nonnegative integers Ai > A2 > . . . > An, summing to n, and denoted by A = [Ai, A 2 , . . . , A^]. This is often called a partition of n as well. To indicate repetitions it is convenient to use notation as we did for conjugacy classes,
A = [n""
...2''n''']
for flj repetitions of j , satisfying YlJ^j ~ "• The shape is represented by an array of boxes, with row j consisting of A^ boxes. E.g., A = [54221] is a partition of n = 14 and is the shape
A standard or Young tableau is a Young frame filled with the integers 1 , 2 , . . . , n so that the numbers are strictly increasing along each row and down each column. For example, 1 3 4 6 7 2 5 8 10 9 12 11 13 14 is a standard tableau of the above shape. Many properties of Young tableaux depend on the notion of hooks. 1.2.1 Definition. The hook corresponding to a cell in a Young frame is the cell itself plus the cells lying below and to its right. In the above tableau, e.g., 5, 8, 10, 12, 13 define the hook of 5 having length 5. A beautiful result of Frame, Robinson and Thrall counts the number of standard tableaux of a given shape.
114
CHAPTERS
1.2.2 T h e o r e m .
Frame-Robinson-Thrall
The number of Young tableaux product of the hook lengths.
formula
of a specified shape is given by n\ divided
by the
The importance of this formula is that, as we will see below, this is the degree dx of the irreducible representation of 5„ corresponding to the shape A. For the shape [322] we have dx =
1-4-5-2-3-1-2
An interesting discussion of tableaux and a probabilistic proof of this formula may be found in the article by Greene, Nijenhuis, ajid Wilf [42]. 1.2.1 Young
symmetrizers
To each standard tableau corresponds the projection in the group algebra T{S„) that maps to the irreducible representation corresponding to that tableau. As we saw in Chapter 4, for a given shape A there are dx copies of the corresponding irreducible representation in the regular representation of Sn on F. The projections ex in the group algebra are the sum over all standard tableau of the corresponding Young symmetrizers. To a given row of a standard tableau corresponds the subgroup of S„ acting as permutations of the symbols, numbers, of that row and fixing everything else. For a row, we consider the symmetric representation of that subgroup, and sum over the corresponding subgroup. For a column, we consider the antisymmetric representation of the subgroup permuting the symbols of the column and sum over the corresponding subgroup. Note that the subgroups acting on different rows permute different elements and hence commute. Similarly for the columns. 1.2.1.1 D e f i n i t i o n . For a given standard tableau, the Young symmetrizer is the element in the group algebra given by QP where Q is the product of the antisymmetric representations of the columns and P is the product of the symmetric representations of the rows. For example, for the tableau of shape [22]
1 3 2 4 we have QP = ((1 - (12))(1 - (34))) ((1 + (13))(1 + (24))) The hook formula gives dx = 2, with the other tableau given by exchanging 2 and 3 in the one above.
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION 1.3
115
SYMMETRY CLASSES OF TENSORS
Schur and Weyl developed the connection between the representations of the symmetric group and representations of GL(N), the general linear group oi N xN invertible matrices (here we are interested only with real or complex entries). Given a vector space V, the nfold tensor product V®" consists of linear combinations of formal products Vi <8iV2 ®- • •®v„ of vectors which is multilinear, i.e., scalars factor out, and the distributive law holds, i.e., v ® ( x + y) = v ® x + v(8)y R e m a r k . Sometimes it is convenient to drop the tensor product signs and think of products of noncommuting variables or concatenation, as in multiplying words of a formal language. If { e i , 6 2 , . . . , e ^ } is a basis for V then a basis for V®" is given by e>i.2...i„ = e.-, ® ei, ® . . . ® e,„ corresponding to N choices per each of n positions. Thus, dim V®" = iV". An element of S„ acts on a tensor by permuting the positions of the vectors. For convenience, as remarked above, we will drop tensor signs, remembering that the order of the symbols is important. E.g., (23)61646565 = 6 1 6 5 6 4 6 5
l.S.l
Irreducible representations
of the symmetric
group
Applying the Young symmetrizer corresponding to a shape A to the tensor Ci ®62 (8>- • - ®eA? gives a vector that can be used to generate the corresponding irreducible representation. By acting on this vector with elements of SN we have a basis for the representation as soon as we have dx linearly independent vectors. Each standard tableau generates an equivalent representation. Example. For [21], use the symmetrizer (1 — (13))(1 + (12)) corresponding to the tableau
1 2 3 We write just abc for the tensor ei ® 62 ® 63. Thus, Vi = abc + bac — cba — cab V2 = (23)vi = acb + bca — cab — cba With this as basis, one checks the representation
(0
Thus, the characters Xp J211 _ 0^>
X(21) -
this last being given by the degree.
Y''•1) A( 3) -
-1
yt^^i - 2
1
116 1.S.2 Characters of GL(N).
CHAPTERS
S-functions,
Given a basis for V, a linear transformation A acts as a matrix (a,j) with Acj = ^ Oije;. This extends naturally to tensors. E.g., Acj ® e* =
^^aijoik^i
le;
',1
That is, A effectively acts on each factor independently. We want to calculate the character of this action of A, i.e., the trace. By a similarity transformation, we may assume A is upper triangular with the eigenvalues, say, xi,... x^ on the diagonal. Acting on n-tensors, contributions to the trace will consist of monomials in the Xi. For each shape A, applying the Young symmetrizer corresponding to a given standard tableau to an n-tensor gives a tensor of syraraetry CISLSS A. Since repetitions are allowed in any given basis element of V®", one considers column-strict tableaux where along rows only nondecreasing numbering is required. Each such tableau corresponds to a given basis tensor in the symmetry class and the contribution to the trace is given by the corresponding monomial in the variables x,. The sum of these monomials gives the character of the representation of GL{N) and is denoted {A}, the Schur function or S-function . To illustrate: Example. Consider the action of a 2 x 2 matrix on V®^ corresponding to the shape [32]. Then the ^-function {32} = xlx2 + x^x^ corresponding to the diagrams
1 1 1 2 2
1 1 2 2 2
in turn corresponding to the number of repetitions of the tensors ei and 62 respectively. Two particularly important classes are the symmetric tensors and the antisymmetric tensors . The symmetric tensors are gotten by symmetrizing over all positions. On ntensors, this action is the Young symmetrizer for the shape [n]. Similarly, antisymmetrizing over all indices, i.e., putting in factors acccording to the parity of the permutations, is the Young symmetrizer for shape [1"]. The corresponding representations of GL(N) are respectively the fully symmetric and antisymmetric representations. The corresponding S-functions are respectively the homogeneous symmetric functions and the elementary symmetric functions . That is, 1.3.2.1 P r o p o s i t i o n . The S-function {n} is the homogeneous symmetric function consisting of the sum over all monomials homogeneous of degree n in the variables Xi,... ,XNThe S-function {!"} is the elementary symmetric function consisting of the sum of all monomials Xi^ • • • Xi^ corresponding to all choices of n distinct indices.
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
117
1.3.2.2 Definition. We denote by /i„ and a„ the n'** homogeneous and elementary symmetric functions respectively, in a given set of variables xi,... ,XN- We denote the power sum symmetric functions by
5. = E^" i=l
Remark. Note that if A is an A'^ x Af matrix with eigenvalues xi,... sum Sn = t r ^ " .
^ XN, then the power
There are a variety of formulas for 5-functions. We note without proof: 1.3.2.3 P r o p o s i t i o n .
For a given shape X the S-function r^i ^ ^
{ A } may be expressed as
det(xf>+^-^) det(rf-^) = det(/iAi+j-i)
'
'
p
with det(x; ~') the Vandermonde determinant in the variables xi,... , XN. The equality of the Brst two hnes is the Jacobi-TVudi identity and the last formula noted is the FrobeniusSchur formula expressing the S-function { A } as t i e Fourier transform of monomieds in the power sums. In these formulas, ho = UQ = 1 and /i„ = a„ = 0 if n < 0. In particular, since h„ = {n} and On = {!"}, corresponding to the trivial and alternating representations respectively, we have
p
1
Y^
We will see some more formulas for 5-functions later on in the Chapter. 1.4
SOME APPLICATIONS
Here we discuss MacMahon's Master Theorem and Molien's Theorem. We present some simple examples to show the basic ideas behind the theorems.
118
CHAFHERS
1.4.1 The symmetric
representation
and MacMahon's
Theorem
The space of symmetric tensors is effectively the same as polynomials in N variables, since the factors can be treated as commutative variables. Thus, computations on the space of symmetric tensors is not difficult. So taJce as variables ^1, • - • ,^N- And consider polynomials in these variables. The symmetric representation of a matrix is the action on polynomials induced by:
where
where we are actually working with the tensor components, as usually done when working with vectors. Now it is easy to see directly that if A is upper triangular, with eigenvalues xi,... ,XN, then the trace of this action on the space of polynomials homogeneous of degree n is exactly h„(xi,..., XN), the n'^ homogeneous symmetric function. We recall, as are readily proved, 1.4.1.1 P r o p o s i t i o n . As generating functions for the elementary symmetric functions, we have
J J ( 1 +vxi)
=
i=l
and
homogeneous
^v"a„{xi,...,XN) n
N
J J ( 1 - vxi)~^
=
'^v"h„{xi,..
.,XN)
Thus, 1.4.1.2 T h e o r e m .
MacMahon's
Master
Theorem
Denoting by tr g the trace of the symmetric geneous of degree n, we have 1
det(/-tA)
representation
°°
=T.^"^'Syn.A
on polynomials
homo-
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION Proof:
119
We have, with Xi denoting the eigenvedues of A,
=n-^
I d e t ( / - tA)
A l 1 _ fxi
n=0 oo
Tl = 0
as seen above.
•
This provides a powerful way of deriving many identities. Example. Fibonacci numbers are determined by the recurrence F„+i = F„ + F„-i with Fo = I, Fi = I. It is readily checked that the generating function is 1
°° 11=0
Observing that 1 - 1 - i^2 = d e t ( / - tA) for
we compute, with yi = ^2, y2 = (,i + (.2, the monomials
.= 0 ^ '
/
Contributions to the trace come from the coefficient of £"'^"% namely ( a slight change of notation 1.4.1.3 P r o p o s i t i o n .
The Fibonacci numbers
a-ffr=Ti
satisfy
). Thus, with
120
CHAPTER 5
Proof: On the polynomials homogeneous of degree n, we require that n\ -\- n2 — n. The rest follows from the above discussion and MacMahon's Theorem. • It is clear that a similar approach works for any recurrence with constant coefficients, choosing for A a companion matrix, i.e., one with ones on the superdiagonal, except for the last row, which contains the coefficients of the recurrence. Example.
The Tchebychev polynomials of the second kind have the generating function "
1 l-2st
+ <2
Y^i^Unis) n=0
The matrix A is given by
We have j/i = ^2> J/2 = —^i + 25^2- Applying MacMahon's Theorem as in the previous example, we find
Un{s) = 1.4-2 Invariants
and Molien's
j:(^''-y-in2sr
Theorem
Now we consider any matrix representation, say of degree N, of a finite group. An element g is represented as an A'^ x iV matrix, say A{g), and acts on the space of polynomials via the symmetric representation. A polynomial p{x) is invariant under the group if it is unchanged by this action, i.e. p{gx) = p{x) for all g G G, where gx denotes the variables X transformed according to the matrix A{g). The homogeneous invariant polynomials are a subspace of the space of homogeneous polynomials of given degree, a finite-dimensional space. Thus, 1.4.2.1 T h e o r e m .
Molien's
Theorem
Let 6n denote the dimension of the invariant polynomials homogeneous of degree n under the group G. Let A{g) denote the matrices of a representation of G. Then we have the generating function
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
121
Proof: We know from MacMahon's Theorem that this is 1/|G| times the sum over G of the trace of the symmetric representation. Recall from our discussion of the group algebra in Chapter 4 that the sum ^ ^ g is the projection onto the trivial representation in the group algebra. Alternatively, think of decomposing into irreducible representations, then the Schur orthogonality relations imply that summing any character of an irreducible representation yields zero except for the trivial representation, when the result is |G|. So summing any character over all of the group yields \G\ times the multiplicity of the trivial representation in the given representation. In the present context, the trivial representations are just the invariant polynomials. • Let's look at some examples. Example. For the cyclic group of order 3, we have, as we saw in our study of representations of cyclic groups in Chapter 4, two elements x and x^ with characteristic polynomial det(/— tx) = det(J — ix'^) = 1 — <*, where we take for x the matrix
The 3 x 3 identity matrix has characteristic polynomial (1—<)^. Thus we have the generating function V>nc 2/3 1/3
n=0
n=0
For n = 3, we have, in the variables ^i,^2)i^3i as invariants: as — ^i^2^3, the homogeneous symmetric function /^3, the power sum (I + Q + (.1 and the Vandermonde determinant
(6-6X6-6X6-6). Example. For the symmetric group S3, we use the 3 x 3 matrix representation of the action as permutations on three symbols. The 3-cycles have characteristic polynomial 1 — t^ as in the previous example. The transpositions have characteristic polynomial (1 — <)(1 — t'^) as they fix one symbol and exchange the other two. With the identity this gives V/"/J A> ""•
1/3 1/2 1/6 1 _ <3 -f- (1 _ i)(l _ ^2) -f (1 _ i)3
Here the invariants are symmetric polynomials. To expand the right-hand side, we observe the identity 1 1 / 1 1
il-t){l-t^)
2 \l-t^
+ (1-t)
122
II.
CHAPTERS
Clustering distributions
Consider a process where an operation consists of creating a new file and adding an item either to one of the previously existing files or to the new file. The state of the system is described by the number of items in the files. After N operations, there are N files and N items. The occupancy numbers, i.e., the number of items in each file, give a partition of N and the system can be represented by a Young frame. The branciingruiefor representations of the symmetric group says, in particular, that the degrees dx of the irreducible representations satisfy
dx = 22 ^1^ where, for A a partition of N, the sum is over partitions oi N — 1 such that the shape A comes from a shape fi by adjoining a single box at the end of a row or column. We define evolution functions 4>x by the equation X = dx^ 2_/ '^A''?^*«?^A\^ ti—*X
where the factors 4>x\ii correspond to the transition fi —* X. For example, for iV = 8, A = [3221] comes from [2H], [321^], and [32^]. The hook formula verifies the relation among the degrees: ^A = 70 = 14 + 35 + 21. The transition [3211] -^ [3221] is indicated by the diagram
X
2.1 Definition. A generalized Plancherel measuTe on shapes of size N by
on the symmetric groups is defined
p{AN = \) = dl\<|>x\V^r where Af is the normalization factor so that the probabilities add up to 1.
2.2 T h e o r e m .
For a class function f(p) on 5jv, we have the Plancherel
A
p
identity
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
Proof:
123
Using the Schur orthogonality relations we have
X,p,eT
Consequently, 2.3 P r o p o s i t i o n . If d\(j>x = /(A) for some class function f, then the normalizing constant in the corresponding generalized Plancherel measure is the average value of P over the group. R e m a r k . In particular, if f{p) is 1 on the identity and 0 otherwise, we have /(A) = d\/N\, the value of xx at the identity giving dx. The Plancherel identity is then the same as the formula ^d\ = \G\ for the degrees of the irreducible representations of a finite group G. If we define P(A. = A ) = | ^ then we have what is usually called the Plancherel measure on the shapes. This is the case where (fix = 1 for all A. These Plancherel measures have been studied by Versik-Kerov[82] and have come up in work of Logan-Shepp [57]. We will determine x as functions of a parameter q so that we can model clustering phenomena. First we discuss some more features of symmetric functions and representations of the symmetric group. 2.1
CONJUGATE REPRESENTATIONS AND RELATIONS
For each irreducible representation A, with character x , there is A, the conjugate representation corresponding to the shape A which is obtained as the transpose of A, i.e., rows and columns are interchanged. E.g., if A = [5422], then A = [44221]. The conjugate representation is the product of the representation A and the alternating representation, i.e., multiplying by the parity. Thus Xp =sgn(/>)Xp Note that the degrees are the same. Just as hji — {'^l ^n,d Qn — {1"^} ^^ »5*~functions, corresponding to conjugate shapes, we have in addition to the formulas from Prop. 1.3.2.3
CHAPTERS
124 2.1.1 P r o p o s i t i o n .
The S-function
{A} satisfies
{A} =
det{hxi+j-i)
where A is the shape conjugate to A. R e m a r k . An important feature to note is that the relationships connecting the /j's, a's, a's, and {A}'s hold regardless of the existence of underlying variables that make these symmetric functions. The determinantal relations as well as Frobenius-Schur formula only depend on the following generating function relations, given a formal series L{t) that is the generating function for the variables corresponding to the power sums:
n=l CXI
n=0 oo n=0
By convention, ho = UQ = 1, while h„ and a„ vanish for n < 0. 2.1,1 Frobenius
notation
For a shape, the Durfee square is the largest square formed starting from the upper left corner. It is easily seen that this is an 5 x s square with side .s = # { A;: A; > j }. The Frobenius notation for A is written A = ( a i a 2 . ..as\hih2
. • .h,)
where, moving along the principal diagonal, the ai count the number of boxes to the right, including the diagonal box, and similarly, the 6; count the number of boxes below and including the i^^ diagonal box. For A = [5422],
we see that 5 = 2 and A = (53|43). Observe that in this notation, the conjugate shape is (6i . . . hs\ai ... as). In general. Ui = Xi - i + 1,
bj = \j - j + 1
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION 2.2
125
CLUSTERING FUNCTIONS
Now we fix a parameter q and define corresponding functions <j). With ijio = 1, we have for integer n > 0 n—1
n—1
't>n^\{{l+3q),
4>n=X{{l-iq)
>=0
j=0
And for shapes // = [^i,/i2, • • -, A'N], •i^CM) = n
'^''•-
Finally, for a shape A = (0102 . . . as|6i62 • • • 6s), we let "/lA = J l 4>ai • \ \ 4>bi i
i
2.2.1 D e f i n i t i o n . For a conjugacy class p denote the number of non-zero parts by r{p), i.e., if p = (n"" . . . 2^n>''), then r{p) = J^PjWe define the class function F{p) = q^-'-^"^ 2.2.2 L e m m a .
Proof:
We i a v e the
relations
We set the power sum variables to be 5„ — q"~^. Then Pl <,P2 . . . „PN _
„N-r{p)
which is F{p). We have then to check the generating function relations connecting hn and a„ with the variables s„. We have for the s„: 00 Z—rf
Ti=0
ft
n=0
I
This clearly equals exp(L(<)) as required. The result for the a^ follows by mapping q —» —q for the conjugate ^n•
126
CHAPTER 5
2.2.3 L e m m a .
The expansion of the determinajit,
for X a partition of N
F{X) :^ det((t>x,-i+j) involves terms ^(^) such tha.t /j is a partition
of N.
Proof: In the expansion, the column indices, j , are some permutation of corresponding row indices, i. The subscripts on (j) in any given summand are of the form A; — i + j , , summing to
^{\.-i
2.2.4 L e m m a .
+ ji) = J2X. = N
For any shape X, 4>\ divides
F(X).
Proof: Think of F{X) given as the determinant in the previous Lemma. The first column of the determinant is
\
•• J
where there are either i/io's or O's below row s. Since the subscripts increase across each row, the determinant clearly has a factor of Y[ ^a; • Similarly, consider the conjugate shape and replace (f> by . In that form there is clearly a factor of Yl
As a function ofq, for X a partition
of N with Durfee square of side s,
deg{F(X)) = N - s = deg<j>x
Proof:
We have deg<^„ = n — 1 so that deg ^A = 5 ] ( ° ' - 1) + I ] ( ^ ' - l ) = iV + 5 - 2 5 = J V - s
since ^ ( a ; + bj) = N + s. By the previous Lemma, then, deg F{X) > N ~ s. We will show that the maximal degree of any 4'{n) in the expansion of the determinant is no greater than N -s. In det(0A(-i+j), row i has a 4>Q entry just in case A; < i, in which case ijio occurs in column i — A;. Note that if, say, ii > iii then i\ — A,, = Z2 — A^^ implies 0 < ii — 12 =
YOXJNG TABLEAUX AND COMBINATORIAL ENUMERATION
127
Ail "^'2 ^ Oj 8, contradiction. That is, the o's in different rows occur in different columns. The degree of a non-zero product (^(^) is just A'' minus the number of factors having nonzero subscripts, since // is a partition of N and each non-zero fXj contributes a factor f,. of degree /ij — 1. Thus, the terms of maximal degree have the most factors of <{>o. Since s = ^{ Xf. Xi > i], the number of rows containing o equals #{Ai:Ai < 2} = 6i - 5 as b\ is the number of rows of A, i.e., Aj. So, a non-zero 0/^) has 6i factors in all, with at most hi — s factors of <j>o, i.e., at least s factors (/>^j with m / 0. Hence the result. • 2.2.6 L e m m a .
If g = 0, then F{X) =
dx/N\.
Proof: If g = 0, then to be nonzero, r{p) — N, i.e., p is the identity. The result follows a.s noted by the Remark following Prop. 2.3. • Now we have the main theorem 2.2.7 T h e o r e m .
F{X) = dxx/N\.
Proof: By the Lemmas preceding Lemma 2.2.6, we have that F(X) = cx(j>x for some constant ex, independent of q. Evaluating at 5 = 0, Lemma 2.2.6 completes the proof.
• Combining this with the Plancherel identity, we have 2.2.8 T h e o r e m .
For any real q, the measure on shapes P ( A = A) =
dll
is a probafaiJitj distribution on the set of partitions
of N.
Proof: We just have to check the average value of F^ over the group. This is clearly • the same as F([iV]) with q^ replacing q. The result then follows from Lemma 2.2.2. 2.3
SOME PROPERTIES
The motivation behind the terminology now is clear. Namely, if 5 = 1, then P (A = [N]) = 1, since ^„ = 0 for n > 0. This means that only one file contains all of the items — perfect clustering. While if g = —1, then P ( A = [1^]) = 1, since then cj>„ = 0 for n > 0. Here each file accepts only one item — perfect dispersion. And, as previously noted, q = 0 is the standard Plancherel measure. Another observation:
128
CHAPTERS
2.3.1 P r o p o s i t i o n . If q = 1/m, for a positive integer m, then P(there are < m nonempty £les)—l.
Proof:
In this case, i^„ vanishes for n > m.
•
Example. Take q = 1/2. T h e partitions with non-zero probabihty are of the form [JV] or [N-jJ], with 1 < i < N/2. In Frobeniusnotation, f o r ; > 2, [N-j,j] = (N-j,j-l\2,1), with [N -l,l] = (N - j\2) . Thus, 'l'lN-j,j] = N-jj-ih =
4>N-j4>j-\l2
And P(A = [iV])= ^ - ( W
_
N\<j>N{llA)
6(iV + l) (iV + 2)(JV + 3)
iV(iV - 1) ^ • The expected numThe probabihty that there are two nonempty files is j — — T T T T ^ ber of nonempty files, the possibilities being either one or two only, is
(A) = 2^1
-3(^-+A
(iV-f 2)(iV-h3),
Example. Here are some explicit formulas for small values of N. a. For N = 2,
b. For TV = 3, ^^"^-^^J^-6(1+5^X1+2^2) P ( A = [21])=
^(^-^^)^ 6(l+9^)(l+2g2)
P ( A = [111]) =
(l-g)^(l-2g)^ 6(1+52X1+252)
We can consider the limit as 5 —> 00. Note that a shape consisting of a single hook is of the form [TV - j , V\. 2.3.2 P r o p o s i t i o n . single hook shapes.
As 5 —• 00, the measure approaches a uniform distribution
on the
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
129
Proof: O b s e r v e t h a t d e g l = 2(iV - s) a n d deg (/^^(g^) = 2N -2 imply t h a t P (A = A) —• 0 if s > 1. T h u s , as 5 —• oo, t h e m e a s u r e is c o n c e n t r a t e d o n t h e single hooks. For A = [Af — j , I-'], t h e hook formula for d\ yields _
A^!
fN
-1
and P{A = {N-j,V] =
C-')'4'%-.^U ^
'^
m{N-iy.
_ i_ " N
m Finally, we remark that by choosing q to be purely imaginary, we construct a symmetric measure, assigning equal probabilities to conjugate partitions: P ( A = A) = P ( A = A ) =
III.
'^'l^^l'
iV!
Inversion of circulants
Using the theory of 5-functions, we can develop general formulas for the inverses of circulant matrices of arbitrary (fixed) size. Let A be A'^ x A'' with minimal polynomial m-l
am-jx J
p{x) = x"" - ^
of degree m < N. Denote the reciprocal polynomial to p by m
7r(x) = 1 — y^ ajX-' j=i
Define the polynomials 7rjfc(x) = 1 — X^,_i otjX^, ior 0 < k < m. 3.1 P r o p o s i t i o n .
For any < 6 C, the inverse of I — tA
satisfies
m—l
(7 - tA)-'
= •Kit)-'
Yu ^m-X-A^ j=0
which exists whenever n(t) ^ 0.
{tAy
130
CHAPTERS Proof:
Multiplying both sides hy I — tA yields
m—1
m—l
^(i^y'Tr^-i-.W -
^(My+v^_i_;(i)
j=0
j=0 tn—l
m—l
= Y, itAyi^m-l-j(t)
- TTm-jit)) + x ( 0 - t'" J ] « — i ^ '
= 7r(<)
• We can find a formula for any power of A in terms of 5-functions. We associate S-functions t o the polynomial p{x) by considering, with ao = 1, On = (—1)""'"-'Q:„ as elementary symmetric functions in the theory discussed above, as in fact they are the elementary symmetric functions of the eigenvalues of A for p{x) the characteristic polynomial. We associate /i„'s via the generating function 7r(a;)~^ = '^x"h„. 3.2 L e m m a .
The hook S-functions
satisfy r
Proof:
We have via the basic determinant formula a„ = {!"} = det(hi_i+j)
And in the determinant for {n, 1""}, expanding by cofactors along the first row, the minors that arise all have just hi and ho on the diagonal, giving corresponding a;t's. The elements of the first row are of the form hn+k and the indicated result follows. • Combining this with the previous Proposition yields: 3.3 T h e o r e m .
Let ^4 have minimaJ poiynomiaJ p(a;) = a;"* — ^ ^ " ^ a-m-jX^-
Then
m-l
A" = "^{71 - m + 1,1""~^~^} i-lT'''^
A^
Proof: In Prop. 3.1, expand the inverses, using geometric series for the left-hand side and on the right-hand side expanding in terms of the h„. The geometric series will
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
131
converge in a neighborhood of the origin in C for |f | less than the reciprocal of the spectral radius of A. (Optionally, one can consider these as calculations with formal series.) f
Y^t-A- = (Y^t^h^ fc=0
m-\
Y. ^ru-^-,{i){tAy /
\
J=0
tn —1 /
E
}=0
m—J—2
\
k=0
and the result follows from Lemma 3.2.
•
R e m a r k . Note that the above discussion is valid as long as A satisfies p{A) = 0. In particular, for the remainder of this section, we will take p to be the characteristic polynomial of A so that 5r(x) = det( J — xA) Now, the question is how to compute the determinant and adjugate (determinant times the inverse) of f{A) for any polynomial of degree less than m. The idea is to 'transfer' to a companion matrix for / , T, which is independent of A. 3.4 D e f i n i t i o n . matrix
For the polynomial f(x) / 0 0 0 \pi
1 0 0 p,_,
= 1 — Ylj==i Pj^'' •: the associated 0 1 0
... ... ...
/3,_2
companion
0 \ 0 0
A/
This is chosen so that 3.5 P r o p o s i t i o n .
The companion matrix T satisfies
fix) = det(J - xT)
Now recall the Kronecker product of matrices. This is the same as the tensor product of the matrices. If T and A act on V, then A®T acts on Vi ® V2 6 V ® V as {A ® r ) ( v i ® V2) = Avi (gi Tv2 extending to all of V ® V by linearity. Notice that if { f; } and { rjj } denote the eigenvalues of A and T respectively, then the eigenvalues of A ® T are { (iTjj } . So det(7® /— A ® T) = f l ; (1 — i,,rij). Now the idea is to keep A and take the 'partial determinant' with respect to T yielding /(A) = d e t ( / ® / - A ® T ) (3.1) \Ve now have the principal result
132
CHAPTER 5
3.6 T h e o r e m .
Let f(x)
= 1 - Y^'j^^ fijx', I < m.
Then
det / ( A ) = det 7r(T)
adj f{A) = det Y, ^m-i-jil
Proof:
®T){A® Ty
To get the first line, in eq. (3.1), interchange the order of taking determinants: det / ( A ) = det det(J
®I-A®T)
= det det(i"
®I-A0T)
= det 7r(T) Now write Prop. 3.1 replacing A by A ® / and T by / ® T and rearrange to get m—\
7r(/ ® T) = ( / ® / - A ® T) Y^ 7 r ^ - i - j ( / ®T){A®
Tf
3=0
Taking determinants with respect to T yields m—l
d e t ( / ( A ) ) / = / ( A ) X det ^ 7 r „ - i - , ( / ®T)iA® j=o which identifies the latter factor as adj A.
3.1
Ty
•
REDUCED DETERMINANTAL F O R M O F FOULKES
At this point, we need another result on 5-functions. It was noted by Foulkes that any S-function can be expressed simply as a determinant of hook 5-functions. 3.1.1 Definition.
We have, using a modified Frobenius notation, the 5-function
{A} =
di,d2,...
for the partition A = ( n i , r ( 2 , . . . |c(i + 1, c/g + 1,...) in the standard Frobenius notation. The reduced determinant al form is
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
3.1.2 L e m m a .
Foulkes' reduced determinantal
where the entries are S-functions
form is
'ni,n2,.
{A} =
di,d2,...
133
detiiujl-"})
_
for the hook diagrams
indicated.
For example, we have {54211} =
'5,3' 4,1
{51*} {51}
{31*} {31}
and
{54} =
3.2
5,3 1,0
{51} {5}
{31} {3}
CIRCULANTS
For circulants, we have, as in Ch. 4, §2.1.1, A = U satisfying J7" — / = 0, i.e., p(x) = x " — 1, taking U as generator of the cydic group of order n. So 7r(x) = 1 — x" and 7r;t(a^) = 1, for k < n. 3.2.1 P r o p o s i t i o n . matrix for f,
For a circulant f{U) = 1 — Ylj=i PjU', y^ith T the
companion
del f(U) = det(I - T")
adifiU) =det J2iU®Ty J=o
Proof:
This follows directly from Theorem 3.6.
•
R e m a r k . In the following discussion, with f(x) = 1 — 2 , = i l^j^^•> *he corresponding elementary symmetric functions are 6^ = (—l)-'"'"^/?j, for j > 0. I.e., these are the elementary symmetric functions in the eigenvalues of the companion matrix T. 3.2.2 L e m m a . k > I
Let T be the companion matrix for f{x) = 1 — X^,_j 0jX^. Then, for
{T^j^i-iy'^ik-l with the S-functions
corresponding
to the b„.
+ i,!'-^}
134
CHAPTER 5
Proof: The Cayley-Hamilton theorem yields T' = X^j=o Pi-i'^'• Consider the action of T as a matrix on the column vector T = (J, T , . . . , T'~^)', the t denoting transpose — y r ^ (T*, T * + \ . . . , T^+'-i)*. From Theorem 3.3, we have, for r > /, i-i
T'- = ^ { r - / + 1, ! ' - > - ! } ( - l ) ' - J - i T ^ 3=0
Substituting into (T*, r * + ^ . . . , T*+'~^)' for ^ < r < fc + / and matching with the action on T yields the required form. • 3.2.3 T h e o r e m . Let f{x) = 1 — ^ . - i PjX^, with compeuiion matrix T. Then, /t"] = d e t ( / - xT"), we have
Jt=o
o
*, * 1,
, 1
/
where \i\ = ii + 12 H \-ik, and tiae reduced determina.nta.1 form refers to the corresponding to the bk = (—1 )*"'''/3yt. Proof.
setting
S-functions
T h e theorem follows from the standard formula det{I - xR) =
^{-x)''R(k)
for a matrix R, with R{k) the sum over k x k principal minors. Here we have R = T " given explicitly by the above Lemma in terms of hook 5-functions. The minors are all of the indicated reduced determinantal form and one checks carefully the minus signs as needed. • 3.2.4 Corollary.
The determinant
of the circulant f{U) is given by det/(J7)=/M(l)
Example.
One can apply this result to find the following formulae:
a. For f{x) = 1 — ax — /3a;^, we have det f(U) = l-{n}
+ {n-l,l}
+ {nn}
h. For f(x) = 1 — ax — /3x^ — 7 1 ' , det f{U) = 1 - {n} + {n - 1,1} + {nn} - { n - 2 , 1 , 1 } + { n - l , n - 1,2} - {n,n-
1,1} + {nnn}
We are left with the question of computing
adj /([/) = det Y^iU ® TY = n E d U' j=0
k=l
j=0
for the case f{x) = f | ( l — ^jrx). Multiplying out the product for adj f{U), we see that the coefficient of 17'' is the homogeneous polynomial of degree r in the d but with the restriction that no ^t can occur with an exponent greater than n — 1.
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
135
3.2.5 Definition. Given n > 0, we let h'^. denote the truncated homogeneous polynomial which is the homogeneous symmetric function of degree r in given variables, { ^i }, but only including monomials with the property that no variable occurs with an exponent greater than n — 1. So, we write
j=0 k=j
(mod n)
Example. For the case of two variables, we can see from the Young diagrams that h'j._^_^ = {n — 1, k + 1], ioi 0 < k < n — I. As an illustration, ioi n = 6, k — 2, we have h'g = {53} corresponding to the diagrams
1 1 1 2 2 2
X
X
where the x's can be 11, 12, or 22. In general, we have 3.2.6 P r o p o s i t i o n .
Write f''\x)
= 1 - Y!J=I
/ ? ' " ' X J . For 0 < it < n, we have
r
3=1
Proof:
Using geometric series, note that n-l
det J2 ^ ' 2 ^ ' = det [ ( / - x " T " ) ( / -
xT)-']
3=0
oo
= f"\x'')J2x^h, However, the determinant on the left-hand side equals, as we just saw, ^^^^^'rcoefficients of T'^+''" yields the result. •
Equating
Finally, 3.2.7 T h e o r e m .
For a circuiant f{U),
j=0 r=0
Proof: To check the limits of summation over r, observe that, by construction, h'^ = 0 if 5 > (n - 1)/. This says that j + rn < {n - 1)1, oi r < I - (l +j)/n < I. •
136 IV.
CHAPTERS Y o u n g t a b l e a u x a n d combinatorial e n u m e r a t i o n in parallel processing
Here we indicate the connection between Young tableaux and parallel processing. Fran5on[32] proposed recently a quantitative approach to the serializability problem by counting commutation classes, i.e., classes of words that can be derived by commuting some letters in a word. He proved that there is a one-to-one correspondence between certain commutation classes and Young tableaux of a given shape. A basic question in this area is to count the number of Young tableaux with a given number, or at most a given number, of rows. Work of Regev[76], Askey-Regev[7], and D. Gouyou-Beauchamps[43] is of particular interest in this regard. R e m a r k . There are other applications of Young tableaux as well. E.g., the connection between Young tableaux and invariant theory leads to applications in such fields as computer vision. See, e.g., S.S. Abhyankar[l]. Also see Knuth's discussion of Young tableaux, [53], V. 3, p.48fF. Giving exact formulas for the number Y„!c of Young tableaux having n cells and at most k rows is in general an open problem. For k = 2 and k = 3, exact formulae have been given by Regev[76]:
_(2n)!
(2n + l)!
y _vV" V - M
(2n)! where C„ = ———-^—z is the Catalan number and M„ the Motzkin number. D.G. n! (n -h 1)! Beauchamps[43] proved that Y2n-1,4 = C„Cm
5^2n,4 =
C„Cn+l
and ^ Y^^ 3!n!(2» + 2)! ^ in-2i)\il{i + l)l{i + 2)\ii + 3)l "'^
4.1
AN APPLICATION IN PARALLEL PROCESSING
Let X = {xi,X2,...} be a finite alphabet and X* the free monoid generated by X, 1 denoting the empty word. We consider a set of commutation rules on X, i.e. a relation C on X which is reflexive and symmetric. If {xi,Xj) 6 C, we say that the letters Xi and Xj commute and conflict otherwise. 4.1.1 Definition. For w € X*, the commutation class of w denoted by [w] is the equivalence class of w for the congruence generated by {xiXj = XjXi/(xi,Xj) € C}. In other words, the class [w] is the set of words which are generated by application of the derivation rules XiXj —* XjXi for each (xi,Xj) € C.
YOUNG TABLEAUX AND COMBINATORIAL ENUMERATION
137
4.1.2 Definition. Let w and w' be two words of X*, the shuffle product of w and w' denoted by w liAw' is the set defined by: if iz) = 1 then w LUio' = w', ii w' = 1 then w \JJw' = w, In general, let a S X (resp. a' € X) and u € X* (resp. M' € X*) such that w = au (resp.tii' = a'u'), then U! LUw' = a{u [X\w') U a'(u) LLIu') It is not difficult to show that 4.1.3 P r o p o s i t i o n . For the shuSie product, we have associativity, distributivity with respect to union. 1 is the neutral element.
commutativity,
and
The shuffle product of two languages L and L' is
LlUL'=
(J
(u)iUJw2)
4.1.4 L e m m a . Let A, A ' be two disjoint alphabets and w € A* (resp. length n (resp.n'), then : (.n + n')\ , card w LUw = j — - ^
w' € A'*) of
R e m a r k . The problem of analyzing the evolution of two stacks in a common memory area has been solved by Flajolet[25] using shuffles of languages corresponding to one-dimensional random walks . We now consider the application of Young tableaux to the enumeration of some commutation classes. These results of this section are due to Fran5on[32]. Consider an alphabet X = {x\,Xi,X2,X2, • •. ,x„,x„} and the following commutation rules: the only letters which do not commute are Xi and Xj, and Xi, Xj, 1 < i < n, 1 < j < n. For the commutation class C„ = [xiX2 ... a;„;riX2 • • • x„], we get card(Cn) = 1 • 3 • 5 • • • {2n — 1). As shown by Fran5on, this example has an interpretation in queuing theory. Consider a queue where only two kinds of events can happen : i) the arrival of a client, denoted by Xi for the i"' client, ii) the departure of a client, denoted by Xi for the i"' client The behavior of the queue is a sequence of such events, two events never being realized at the same time. If the departure is determined by some priority, such as first-in-firstout or last-in-first-out, we get a priority queue and each element of Cn corresponds to a possible behavior of the priority queue. Frangon found an interesting link between some commutation classes and Young tableaux.
138
CHAPTER 5
4.1.5 P r o p o s i t i o n . Let Cn,m be the commutation class [xiX2 • • • XnXiX2 • • • Xm], for 0 < m < n, with the following commutation rules: the only non-commuting letters are Xi and Xj, Xj and Xj, a:, and xi for i ^ j , i = 1 , 2 , . . . , n and j = 1 , 2 , . . . , m . Then C„^rn is in one-one correspondence with the set of two-rowed Young tableaux, having the first row of length n, the second of length m. In addition, card (C„,m) = — — — — C^+m The previous result can be generalized: 4.1.6 P r o p o s i t i o n . Consider an alphabet {x\ jX^ .,•.. ,x\ ,^2 ,. • • } with commutation rules defined as follows: x\''^ and a;^" commute if and only if i ^ k and j ^ /. Let ni,n2, •••,nk be k integers such that ni > n2 > ... > nk > 0. Then the commutation class J. r (1) (1)
(1) (2) (2)
(2) (3)
(it) (*:)
((t), .
.
oi \x\ x\ . . . Xn(x.\ x\ . . . Xn^x\ x\ x\ . . . i n t J Js m one-one with the set of Young fabieaux of shape [ni, n2,..., nk]-
,
correspondence
Young tableaux thus have interesting applications in performance analysis, since in some cases we can give explicit formulas for the number of elements of the commutation classes. This permits the comparison of protocols, etc. R e m a r k . The work of Viennot[83] is of interest in connecting Lie algebras with formal languages.
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