the differential equation · describing elements of a random graph chernoff bounds too consider a...
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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
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