The differential equationmethod

Laura Eslava.

11 MAS - UNAM

BUC - Chile

Describing elements of a random graph Chernoff bounds tooConsider a graph G on n vertices where

Ipcxza ) =p ( etxzeta ) s IE fetx )edges are present independently at random -

etaVertex set [ n ] = 41,2, .

. . . ,n }-

Edges r.ve 's Xij -

. the.

. - j , independentt=4a/Elai - ai I

'

optimize t

Some

interesting

statistics : If Sn = FIX; independent Xi- number of edges

- isolated vertices lpcsns.EE te ) E Ete,E- Eet ' " # " D)

- number of components C of giver size ) t bounded Xi so that a ;

E Xi E bi

- degree sequence pls.

? ELS . ) + e) e e- ZETEC bi - ai "

- chromatic number

- independence number This gives quantitative bounds for

L LN PC Xi ECO , 17 )

Intuitively changing one variable in txijl

zipI

EIHT> u )

EZ2e

- n°42 - o

N ? In ? Iwont charge statistics by much

.

• , ⇒ PCIE - # Htt > u infinitely offer ) =o

This course : conditions to see

trends . . . in a ( graph ) discrete process ¥-E# o

Two generalization ② Let 3%7 independent X ; EX

① If ttilizo martingale ,-20=0

f :X"

- IR

I Zit ,

- Zilla .

Conditions

PIC -2ns Efate ) E e-

2%92Iflx , ,x , . . .

.in/-fCy.....,ynl/EZcilkxi=yiz

Proof Sketch : Fi -- e - algebra ( Z - - -

Zi )ppg z # I y + e) E e-

2%92

Zn= Ii Zi - Zi . , - ?? V ;

FH ) -

f

E Let-24

-. Ef

.

etui ] Vi -

- Elf Ifi ) - Eff IF :D

=E[I etui # ( etvnlfn. , ] Fi = e - aye ( X

, . . . . Xi )

Now Vn satisfies conditions of Fru . 's Li ,U ;

measurable wit ti ,

Hoeffdirg 's lemma. Liev ; EU ; Ui - Li Eci

so can apply Hoeffding's lemma

Extension in Hamming distance Problem : Eld ( X. At ) unknown

f- ( X, , . . . . ,Xn ) E S

"

ACS"

ENZBZ# KCIA )

Hamming dist d #C x. y )=ZHy×;±y ,

PCdlx.AIS.cn/ElP(dHAHEf.Jtu)dHlx.Al=ifefadHCx.y

) u= Eh - B

[ A ]r=4 IES"

:D#

C x. At Er }Trick : from Hoeffding lemma

'

proof

Ef etlf - Effi ) ) ⇐ entyoo

How big becomes CATen f= - d( X. A )

E [ e-that } e-

TEH ]

FPCKELA )

, )=P( DCI At E En ) PCA )=Efh, etf) EEI

etff-Efe-tdlx.tt/--l-lP(dCX,A) > en )

* En > E[ dck.AMa en . II fdcx.at) PCA )

e-tttdcx.AMsent 't

€ Take logarithm t optimize t

IPCDH.Alsent-IPCDH.AT > Efm ]tu ) se' 2n "

-

II I dlx.AT/sV-nlog-PCAI2

=D

Examplesof analysis of algorithms

① Matchings in Gn, %

C Erdos - Ringit

Graph on n vertices edges independently present up . I

Matching :set of edges that do not share common vertices

Algorithm :

① Delete isolated vertices

② Select a uniformly random vertex of min - degree r

③ Include in matching one of its edges

④ Delete v and all other incident edges .

The :c> o

Thealgorithm obtains

a maximummatching a. a.

s.

( IPG ) - I n - o )

② 3 - coloring regular Gn.

d

uniformly chosen graph on those with n vertices and degCv ) =D all v.

Proper coloring : assign colors to vertices so that no two

adjacent vertices have same color

Algorithm : Classify uncolored vertices according to # available color

Si C t ) = Jv : I possible colors forv1

while So HI -- o

① Select minimum i with Si HI so

② select uniform vertex in Si HI

③ Color v randomly,update Silt ti )

.

Failure if So I t ) so

The : d a 4.003.

The

algorithmobtains a proper 3 - cot. a. a. s

.

③ 3 - SAT formulae

( I,

x X,

x Xz ) n ( XI x I ,

X X, )

↳ literal : t or -

variable

O x I = I

O n I = 0

Assign valves to variables,

to satisfy formula

Pure literal :when formula only contains literals xi ( or only Ii )

Algorithm : while Ipure

literal available

① choose one at random

② assign' positive

' value =! ! x ;

= o ,

③ remove all clauses containing x i

③ 3- SAT formulae

Pure literal :when formula only contains literals Xi ( or only Ii )

Algorithm : while Ipure

literal available

① choose one at random

② assign' positive

' value =! x ;=o ,

③ remove all clauses containing xi

3 literals perclause In : F do > o

.

Construct a random

m clause 3- SAT where #literals = Xi )spoils ; )

n variables and density is d :

density men

dado formula is satisf. a. a. s

.

d > do formula is not satisf. a. a. s

.

An introductory example

There are n bins and men balls,sequentially place balls in depend . into

,

How many empty bins there are left ?-

B

D= .IE/3jBj=Hyj.thb;n;senptyzlPCBj=D=G-Ynlcn=ElB7--ne-Titan)

IFTX ; } iid.

X ; n Uni f En ] → X ; -

- j if i . th ball goes to j - th bin

Now B satisfies the bounded differences condition

DAB - ECB ] HE n ) E 2 e- 25%312=2 e-

2h %

If need to estimate ECB ]

Yi = # empty bins when i balls have been located

I Titi - Yi I E I

If need to estimate ECB ]

Yi = # empty bins when i balls have beer located

Local charges , given current information Fi = generated by" info "

at time i

Eftiti - Yi Ifi ]= -1119 bathto ) =

Idealized ODE :

Intla ylt ) Y = yto ) =L DY .

= Titi - Yi

ELYYIIF.in#=yYtt=-yHlSolution gate

' t ELBE II I Yen ] = e- c

- .

A toy deterministic question

How much car two collections

offnations can differ if they

have similar derivatives and initial values ?

IX. or so small perturbations such that

yfot-y.degiftf-Felt,yet)

I Hot-ghost⇐ xIzfft) -Felt,ZHI) Iso

Condition : F is L - Lipschitz cont.

in norm It . Hoo

in particular I Fit , ya ) - f

ftp.HD/ELmax/yHI-zAll

k k KEI kk

⇒ max I y HI - Htt Iscat art ) e"

ft ET

het k k

ly's ) - E G) k ly ' ( s ) - Fls , #s ) ) I t IFC1¥71

< I Ffs ,yls) ) - FCS, ZGDI + or

E L . !yes) - ZCSH to

light- z HIIElylo) - z CosItfotlyllsl -zilslds¥ EXt ST ) t got Lxlslds

Lt

By Gionwall 's ineq . ⇒ XHI⇐

latest ) e