Mellin Transform Asymptotics

97 Pages·2007·0.72 MB·English

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE
The Average Case Analysis of
Algorithms:
Mellin Transform Asymptotics
Philippe FLAJOLET, Robert SEDGEWICK
N(cid:23) 2956
Aou^t 1996
PROGRAMME 2
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THE AVERAGE CASE ANALYSIS OF
ALGORITHMS:
Mellin Transform Asymptotics
1 2
Philippe Flajolet & Robert Sedgewick
Abstract. This report is part of a series whose aim is to present in a
synthetic way the major methods of \analytic combinatorics" needed in the
average{case analysis of algorithms. It reviews the use of Mellin-Perronfor-
mul(cid:26) and of Mellin transforms in this context. Applications include: divide-
and-conquer recurrences, maxima (cid:12)nding, mergesort, digital trees and plane
trees.
L'ANALYSE EN MOYENNE D'ALGORITHMES:
La transformation de Mellin
R(cid:19)esum(cid:19)e. Ce rapport fait partie d'une s(cid:19)erie dont le but est de pr(cid:19)esenter de
mani(cid:18)ereuni(cid:12)(cid:19)ee les principales m(cid:19)ethodesde \combinatoire analytique" utiles
(cid:18)a l'analyse d'algorithmes. Il y est d(cid:19)ecrit l'utilisation des formules de Mellin-
Perron et de la transformation de Mellin dans ce contexte. Les applications
comprennent: les r(cid:19)ecurrences diviser-pour-r(cid:19)egner, la recherche de maxima,
le tri-fusion, ainsi que les arbres digitaux et les arbres plans.
1
Algorithms Project, INRIARocquencourt, F-78153 Le Chesnay (France)
2
Department ofComputerScience, Princeton University, Princeton, New-Jersey08544
(USA)
i
Foreword
This report is part of a projected series whose aim is to present in a
synthetic way the major methods and models in the average{case analysis
of algorithms. It belongs to a collection of reports relative to \Analytic
Combinatorics" that comprises the following sections:
I. Counting and generating functions.
II. Complex asymptotics and generating functions.
III. Saddle point asymptotics.
IV. Mellin transform asymptotics.
V. Functional equations.
VI. Multivariate asymptotics.
Parts I, II, and III consist of 6 chapters that have been issued as INRIA
Research Reports 1888 (116 pages, 1993), 2026 (100 pages, 1993), and 2376
(55 pages, 1994).
The present report constitutes Section IV. It is devoted to complex
asymptotics methods based on Mellin transforms. It consists of one chapter
(numbered consecutively after those of Parts I, II, III):
7. Mellin Transform Asymptotics.
Acknowledgements. The workofPhilippe Flajoletwassupported inpart bythe
Long Term Research Project of the European Union Alcom-IT (# 20244).
The authors are grateful to Tsutomu Kawabata for a careful scrutiny of the
paper and for detailed technical comments.
ii
Chapter 7
Mellin Transform
Asymptotics
Die Theorie der reziproken Funktionen und Integrale
ist ein centrales Gebiet, welches manche anderen Gebiete
der Analysis miteinander verbindet.
| Hjalmar Mellin
This chapter presents a collection of closely related methods for the asymp-
totic analysis of sums that arise in combinatorial problems and have a
number-theoretic (cid:13)avour. Such sums involve coe(cid:14)cients either related to
the multiplicative structure of integers (the number of divisors in the anal-
ysis of the expected height of plane trees), to the binary representation of
integers(thenumberof1-digitsforthebestcaseofthesortingmethodknown
as \mergesort"), or to the powers of 2 (the Bernoulli splitting process and
the analysis of digital trees also known as \tries" in computing applications.
Typically, what is required there is to estimate asymptotically sums of
moreorless standardfunctionsweighted bycoe(cid:14)cients that(cid:13)uctuaterather
wildly, the resulting estimates themselves showing sometimes traces of such
oscillatory behaviour.
The Mellin transform is a classical integral transform closely related to
the Laplace and Fourier transforms. It establishes an explicit mapping be-
tween the asymptotic expansions of a function near zero and in(cid:12)nity on the
one hand, and the set of singularities of the transform in the complex plane
on the other hand. At the same time, it transforms a general class of sums,
1
2 CHAPTER 7. MELLIN TRANSFORM ASYMPTOTICS
called harmonic sums, into a tractable factored form. Regarding applica-
tions in combinatorics and the analysis of algorithms, the power of Mellin
transformmethodsderives in anessential wayfromthe combination ofthese
two features.
There is actually a whole galaxy of methods related toMellin transform.
A simple type that is especially close to basic analytic number theory deals
withtheanalysisofcoe(cid:14)cientsofDirichletseries,andwestartourexposition
with examples falling into this category. An important application is the
exactasymptoticsofdivide-and-conquer recurrencesthatarisein connection
with one ofthe mostfruitful paradigm ofalgorithm design. This is asubject
that we discuss in some detail and Mellin related techniques are especially
instrumental in analyzing thefractalcomponentthatis oftenpresentin such
algorithms.
Mellin transformslargely originate in number theory, going back to Rie-
mann's celebrated memoir on the distribution of prime numbers. They have
found applications in the theory of functions, as initially showed by Mellin
(see [33] for a biographical notice with references), and in various areas of
applied mathematics. As should become apparent in this chapter, Mellin
transform are also part of the arsenal of asymptotic methods for discrete
mathematics and the analysis of algorithms.
This chapter is based on a series ofpapers dealing with Mellin transform
asymptotics in analytic combinatorics [15, 16, 17, 22].
7.1 Dirichlet series and coe(cid:14)cient formulae
Many applications in combinatorial analysis, discrete probability, and the
analysis of algorithms involve quantities of an arithmetic nature. For in-
stance, the analysis of the expected height of plane trees involves the divisor
function d(k) (the number of divisors of integer k), register allocation and
somesortingnetworkslead toquantities related tothe binary representation
of integers, like v2(k) (the exponent of 2 in the prime number decomposition
of integer k) or (cid:23)(k) (the number of 1-digits in the binary representation
of k), etc.
Algebra of Dirichlet series. In situations involving arithmetic quanti-
ties, asymptotic estimates are best performed by using Dirichlet generating
functions rather than ordinary or exponential generating functions. A clear
treatment of the elementary aspects discussed in this section can be found
7.1. DIRICHLET SERIES AND COEFFICIENT FORMULAE 3
in Apostol's book [2].
1
De(cid:12)nition 7.1 Let fangn=1 be a sequenceof complex numbers. The Dirich-
let generating function, DGF in short, of the sequence is the formal sum
X1 an
(cid:11)(s) = :
s
n
n=1
The simplest Dirichlet series is the zeta function of Riemann,
X1 1
(cid:16)(s)= ;
s
n
n=1
where the sum de(cid:12)nes an analytic function of s in the half-plane <(s) > 1.
Other examples of Dirichlet series are
X1 ((cid:0)1)n(cid:0)1 1(cid:0)s X1 1 2(cid:0)s
= (1(cid:0)2 )(cid:16)(s); = ;
s k s (cid:0)s
n (2 ) 1(cid:0)2
n=1 k=1
which de(cid:12)nes the \alternating" zeta function, and the DGF of the charac-
teristic function of powers of 2 for <(s) > 0.
Given three DGFs, (cid:11)(s);(cid:12)(s);(cid:13)(s) with coe(cid:14)cients an;bn;cn, sum and
product relations translate over coe(cid:14)cients as follows,
(cid:11)(s)= (cid:12)(s)+(cid:13)(s) =) an = bn+cn
X
(cid:11)(s)= (cid:12)(s)(cid:1)(cid:13)(s) =) an = bdcn=d;
d j n
where the sum is over the integers d (cid:21) 1 that divide n, a property written
d j n. The relation that corresponds to the product is called the multiplica-
tive convolution or the Dirichlet convolution of coe(cid:14)cients. From it, we see
for instance that
X1 d(n) 2 X1 v2(n) (cid:16)(s)
= (cid:16) (s); = :
s s (cid:0)s
n n 1(cid:0)2
n=1 n=1
In the same vein, one has the famous product formula of Euler for (cid:16)(s):
Y 1
(cid:16)(s) = ; (7:1)
1
p 1(cid:0) ps
4 CHAPTER 7. MELLIN TRANSFORM ASYMPTOTICS
where the product ranges over all primes p. The identity (7.1) is easily
checked by distributing the products in
(cid:18) (cid:19)
Y 1 Y 1 1
= 1+ + +(cid:1)(cid:1)(cid:1) ;
(cid:0)s s 2s
1(cid:0)p p p
p p
and it is logically equivalent to the property that every integer decomposes
uniquely as a product of prime powers.
Aninterestingapplication ofDirichlet convolutionsistothefamousMoe-
bius inversion relations. De(cid:12)ne (cid:22)(n), the Moebius function, by
8
(cid:11)1 (cid:11)r >< ((cid:0)1)r if (cid:11)1 = (cid:1)(cid:1)(cid:1)= (cid:11)r = 1
(cid:22)(p1 (cid:1)(cid:1)(cid:1)pr )=
>: 0 if some (cid:11)j (cid:21) 2.
and (cid:22)(1)= 1. Then
1
X (cid:22)(n) 1
= :
s
n (cid:16)(s)
n=1
P
The relation (cid:11)(s)= (cid:12)(s)(cid:16)(s) is equivalent to an = d j nbd. Solving for
(cid:0)1
bn is achieved by solving for (cid:12)(s) which gives (cid:12)(s) = (cid:11)(s)(cid:16) (s). Expressing
the Dirichlet convolution in turn yields the Moebius inversion relation:
X X n
an = bd =) bn = ad(cid:22)( ):
d
d j n d j n
Moebius inversion is in particular useful in dealing with ordinary gener-
ating functions. The in(cid:12)nite functional equation
X1
d
f(z )= g(z);
d=1
with g(z)= O(z) at z = 0, admits the formal solution
1
X
d
f(z)= (cid:22)(d)g(z );
d=1
asis directly obtained byapplying Moebius inversion tothe induced relation
oncoe(cid:14)cients. Asanapplication, lettheclassG bethemultisetconstruction
(as de(cid:12)ned in Chapter 1) applied to F. Then,
!
1
X 1 m
G(z)= exp F(z ) ;
m
m=1
7.1. DIRICHLET SERIES AND COEFFICIENT FORMULAE 5
or, taking logarithms and applying Moebius inversion on coe(cid:14)cients,
X1 (cid:22)(d) d
F(z)= logG(z ):
d
d=1
This yields an explicit enumeration for the component class F when the
function (OGF) of the multiset class G is known. A typical instance is the
counting according to degree of the class F (cid:26) G = GF(q)[x] of irreducible
(cid:0)1
(monic) polynomials over a (cid:12)nite (cid:12)eld for which G(z) = (1 (cid:0) qz) : the
number of irreducible polynomials of degree n is
qn X n=d
Fn = (cid:22)(d)q :
n
d j n
Admissibility of the unlabelled cycle construction, as stated in Chapter 1,
obeys similar principles.
Exercise1. FindtheDGFsoflog(n),ofp(n)andofp(n)q(logn)with
p;q arbitrary polynomials.
P P
(cid:0)s (cid:0)s
Exercise 2. ForDGFs(cid:11)j(s)= aj;nn and(cid:12)(s) = nbnn that
satisfy (cid:12)(s) =(cid:11)1(s)(cid:11)2(s)(cid:11)3(s), one has
X
bn = a1;n1a2;n2a3;n3:
n1n2n3=n
Exercise 3. The OGFs of d(k) and v2(k) are given by
X1 k X1 zm X1 k X1 z2m
D(z)= d(k)z = 1(cid:0)zm; V2(z)= v2(k)z = 1(cid:0)z2m:
k=1 m=1 k=1 m=1
(cid:0)1 (cid:0)1 (cid:0)
Show that D(z) (cid:24) (1 (cid:0) z) log(1 (cid:0) z) as z ! 1 and V2(z) (cid:24)
(cid:0)1
log2(1(cid:0)z) , but that singularity analysis cannot be applied as the
functions cannot be extended beyond jzj=1.
Exercise4. Provethetranslationoftheunlabelledcycleconstruction
stated in Chapter 1.
0
Exercise 5. Assumingthatg(x)isanalyticat0andg(0)=g (0)=0,
then
1 1
X X
f(nx)=g(x) =) f(x)= (cid:22)(n)g(nx):
n=1 n=1
6 CHAPTER 7. MELLIN TRANSFORM ASYMPTOTICS
X 1
(cid:16)(s)= (<(s)>1) De(cid:12)nition
ns
n(cid:21)1
1(cid:0)s X ((cid:0)1)n(cid:0)1
(1(cid:0)2 )(cid:16)(s) = (<(s) >0) Alternating zeta function
ns
n(cid:21)1
Y
(cid:0)s (cid:0)1
(cid:16)(s)= (1(cid:0)p ) (<(s) >1) Euler product
p
1 X (cid:22)(n)
= (<(s) >1) Moebius function
(cid:16)(s) ns
n(cid:21)1
2m
(2(cid:25)) m(cid:0)1
(cid:16)(2m)= ((cid:0)1) B2m Bernoulli numbers and zetas
2(2m)!
B2m
(cid:16)((cid:0)2m)=0; (cid:16)((cid:0)2m+1)=(cid:0) Bernoulli numbers and zetas
2m
1 0 p
(cid:16)(0)=(cid:0) ; (cid:16) (0)=log 2(cid:25)
2
1(cid:0)s (cid:0)s (cid:25)s
(cid:16)(1(cid:0)s)=2 (cid:25) (cid:0)(s)(cid:16)(s)cos( ) Functional equation (1)
2
s (cid:0)s=2 1(cid:0)s (cid:0)(1(cid:0)s)=2
(cid:0)( )(cid:25) (cid:16)(s) =(cid:0)( )(cid:25) (cid:16)(1(cid:0)s) Functional equation (2)
2 2
1
(cid:16)(s)= +(cid:13)+(cid:13)1(s(cid:0)1)+(cid:1)(cid:1)(cid:1) (s!1) Singular expansion at s=1
s(cid:0)1
1=2(cid:0)<(s)
(cid:16)(s)=O(t ) (=(s)!(cid:6)1;<(s)<0) Growth at (cid:6)i1
Figure 7.1: A summary of the main properties of the zeta function.
Analysis of Dirichlet series. A clear introduction to the analytic prop-
erties of Dirichlet series is given in Chapter IX of Titchmarsh's book [42]
to which we globally refer for this section. It is a standard theorem of the
theory of Dirichlet series that they converge in a half{plane <(s) > (cid:27)c and
converge absolutely in a (possibly smaller) half{plane <(s) > (cid:27)a, where by
elementary analysis 0 (cid:20) (cid:27)a (cid:0)(cid:27)c (cid:20) 1, see [42, p. 290{292]. Thus, Dirichlet
series exist and areanalytic in half-planes. Aswe haveseen, the pair((cid:27)c;(cid:27)a)
is (1;1) for (cid:16)(s). For the alternating zeta function, it is (0;1), for the char-
acteristic function of powers of 2, it is (0;0). The half-plane may be empty
n
as in the DGF of an = 2 for which (cid:27)c = (cid:27)a = +1 or equal to the whole
(cid:0)n
complex plane as for an = 2 for which (cid:27)c = (cid:27)a = (cid:0)1. In full generality,
a Dirichlet series exists analytically in some region only if its coe(cid:14)cients are
polynomially bounded.
Itisaclassicaltheorem[42,44]that(cid:16)(s)admitsananalyticcontinuation:
(cid:16)(s)extends toameromorphic function in thewhole ofC with only asimple