The Method of Moments Example I Example II Example III Counterexample

Alois Panholzer

Description

The Method of Moments Example I Example II Example III Counterexample Some applications of the method of moments in the analysis of algorithms Alois Panholzer Institute of Discrete Mathematics and Geometry Vienna University of Technology Universite de Paris-Nord, 16.2.2010 1 / 64

• union-find-algorithms


• procedure quicksort


• universite de paris-nord


• quicksort input string

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 75
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

hTeMteohdfooMemtnsxEmalpeIxEmalpeIIxEmalpeIIISomeapplicationsofthemethodof
momentsintheanalysisofalgorithms

AloisPanholzer

InstituteofDiscreteMathematicsandGeometry
ViennaUniversityofTechnology
[e-mail]

UniversitedeParis-Nord,16.2.2010

oCnueteraxm1p/el46
hTeMteohdfoMOutline

monestxEmalpeITheMethodofMoments

xEmalpeIIEExampleI
Totaldisplacementinlinearprobinghashing

ExampleII
Subtreevarietiesinrecursivetrees

ExampleIII
Totalcostsof
Union-Find
-algorithms

Counterexample

axpmelIIIoCnueteraxm2p/el46
Teh

Method

fo

Moments

ehT

ExampleI

ExampleII

Method

fo

ExampleIII

Moments

Counterexample

3

/

46

hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Motivation

Average-caseanalysisof
algorithms

procedure
Quicksort(A:array)
...dne

E.g.,
Quicksort
inputstring:random
permutationofsize
n
I
numberofcomparisons
tosortelements

I
numberofrecursivecalls
tosortelements

xEmalpeIIxEmalpeIIIoCnueterxAnalysisofaveragebehaviourof
parametersinrandomstructures

maE.g.,
randombinarysearchtree
of
ezisnI
numberofleavesintree

I
depthof
j
-thsmallestnodein
eert

4p/el46
hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Motivation

Average-caseanalysis:

xEmalpeIIxEmalpeIIIX
n
:parameter(i.e.,randomvariable)underconsiderationfor
randomsize-
n
instance

I
Expectation(=meanvalue)
E
(
X
n
)

I
Concentrationresults,Variance
V
(
X
n
)

I
Limitingdistributionresults

X
n
(
d
−
)
→
X
,
X
n

ocvnreegsniidtsrI
Tailestimates(“boundsonrareevents”)

bituoinot.r.vCXuotnrexema5p/el46
hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Showinglimitingdistributionresults

xEmalpeIIBasis:TheoremofFre´chetandShohat
(Secondcentrallimittheorem)
fI

xEmalpeIIIoCnu(
i
)
allpositive
r
-thintegermomentsof
X
n
convergetothe
r
-th
momentsofar.v.
X
:

E
(
X
nr
)
→
E
(
X
r
)
,
forall
r
≥
1

(
ii
)
thedistributionof
X
isuniquelydefinedbyitsmoments

then
X
n
(
−
d
→
)

X,.i.e,Xn

ocvnreegsniidtsirubitnootXeteraxm6p/el46
hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Showinglimitingdistributionresults

xEmalpeIIxEPmalpeIIIoCnuThismeans:thedistributionfunction
F
n
(
x
)=
{
X
n
≤
x
}
of
X
n
converges
pointwise
forevery
x
∈
R
tothedistributionfunction
F
(
x
)=
P
{
X
≤
x
}
of
X
.

etrnPConsider
X
n
=
i
=1
Y
n
,
i
,
Y
n
,
i
independentidenticallydistr.as
Y
,
P
{
Y
=1
}
=
P
{
Y
=
−
1
}
=
21
.

xema7p/el46
hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Showinglimitingdistributionresults

xEmalpeIIxEPmalpeIIIoCnuThismeans:thedistributionfunction
F
n
(
x
)=
{
X
n
≤
x
}
of
X
n
converges
pointwise
forevery
x
∈
R
tothedistributionfunction
F
(
x
)=
P
{
X
≤
x
}
of
X
.

etrnPConsider
X
n
=
i
=1
Y
n
,
i
,
Y
n
,
i
independentidenticallydistr.as
Y
,
P
{
Y
=1
}
=
P
{
Y
=
−
1
}
=
21
.

:01=n

xema7p/el46
hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Showinglimitingdistributionresults

xEmalpeIIxEPmalpeIIIoCnuThismeans:thedistributionfunction
F
n
(
x
)=
{
X
n
≤
x
}
of
X
n
converges
pointwise
forevery
x
∈
R
tothedistributionfunction
F
(
x
)=
P
{
X
≤
x
}
of
X
.

etrnPConsider
X
n
=
i
=1
Y
n
,
i
,
Y
n
,
i
independentidenticallydistr.as
Y
,
P
{
Y
=1
}
=
P
{
Y
=
−
1
}
=
21
.

:02=n

xema7p/el46
hTeMteohdfooMemtnsxEmalpeITheMethodofMoments

Showinglimitingdistributionresults

xEmalpeIIxEPmalpeIIIoCnuThismeans:thedistributionfunction
F
n
(
x
)=
{
X
n
≤
x
}
of
X
n
converges
pointwise
forevery
x
∈
R
tothedistributionfunction
F
(
x
)=
P
{
X
≤
x
}
of
X
.

etrnPConsider
X
n
=
i
=1
Y
n
,
i
,
Y
n
,
i
independentidenticallydistr.as
Y
,
P
{
Y
=1
}
=
P
{
Y
=
−
1
}
=
21
.

n:04=

xema7p/el46