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The Method ofConditional Probabilities

presented by Kwak, Nam-juApplied Algorithm Laboratory

24 JAN 2010

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Table of Contents• A Starting Example• Generalization• Pessimistic Estimator• Example of Pessimistic Estimator

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A Starting Example• Proposition: For every integer n there

exists a coloring of the edges of the complete graph Kn by two colors so that the total number of monochro-matic copies of K4 is at most . 52

4

n

4

A Starting Example

Kn

K4Is it monochro-matic?

There are K4’s in a Kn.

A K4 is monochromatic with a prob-ability of 2-5.

4n

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A Starting Example• Proposition: For every integer n there

exists a coloring of the edges of the complete graph Kn by two colors so that the total number of monochro-matic copies of K4 is at most . 52

4

n

the expected number of monochromatic copies of K4 in a random 2-edge-coloring of Kn.

This proposition says that a coloring for a Kn exists, such that it has at most monochromatic copies of K4.

524

n

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A Starting Example• Let us color a Kn so that it may have

at most monochromatic K4’s.• Such a coloring can hopefully be

found in polynomial time in terms of n, deterministically.

• RED and BLUE are used for coloring.

524

n

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A Starting Example• K: each copy K4 of Kn

• w(K): given a K4, namely K…– at least 1 edge is colored red and at

least 1 edge is colored blue : w(K)=0– 0 edge is colored: w(K)=2-5

– r edges are colored, all with the same color, where r≥1: w(K)=2r-6

• W=K

Kw )(

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A Starting Example• Coloring strategy– Color each edge of Kn in turn. It will be

finished in n(n-1)/2 stages.– Assume that, at a given stage i, a list of

edges e1, …, ei-1 have already been col-ored.

– Then, we should color ei, right now.

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A Starting Example• Coloring strategy–Wred, Wblue: the value of W after coloring

ei red and blue, respectively.–W=(Wred+Wblue)/2– Color ei red if Wred≤Wblue, blue otherwise.– Then, W never increase for all the

stages.

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A Starting Example• Coloring strategy– Since W is non-increasing and the initial

value is , the final value of W is less than equal to it.

– The final value of W (after coloring all the edges) is the actual # of monochromatic K4’s of Kn.

524

n

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Generalization• A1, …, As: events• • ϵ1, …, ϵq: binary, q stages•

kAs

i i 1]Pr[

]}1,,...,|Pr[],0,,...,|Pr[min{

]1,,...,|Pr[21]0,,...,|Pr[

21

],...,|Pr[

11

111

1

11

111

1

11

1

j

s

iij

s

ii

j

s

iij

s

ii

j

s

ii

AA

AA

A

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Pessimistic Estimator• There are cases for which the previ-

ous approach does not work well.• Under the following 2 conditions,

• We can say

2)1,,...,()0,,...,(

),...,( 1111111

j

ijj

ij

jij

fff

],...,|Pr[),...,( 11 jijij Af

s

iji

s

ij

ij Af

11

11 ],...,|Pr[),...,(

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Example of Pessimistic Estimator

• Theorem: Let be an n by n matrix of reals, where -1≤aij≤1 for all i, j. Then one can find, in polynomial time,ϵ1, …, ϵn∈{-1, 1} such that for every i,1≤i≤n,

njiija 1,)(

)2ln(21

nnan

jijj

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Example of Pessimistic Estimator

• • Ai: the event • α=β/n• • Since ,• We define pessimistic estimators

)2ln(2 nn

n

j ijja1

2)cosh()(

axax eexxG

2/22

)( xexG 2

)()()()( yxGyxGyGxG

n

pjij

p

jijjp

ip aGaGef

111 )(2),...,(

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Example of Pessimistic Estimator

• We should show

• In addition, .• Of course, those claims can be

proved, however, here the proofs are skipped.

2)1,,...,()1,,...,(

),...,( 1111111

p

ipp

ip

pip

fff

],...,|Pr[),...,( 11111 pipip Af

n

i

if1

0 1

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Conclusion• Here, we learnt a way to extract de-

terministic information from random-ized approaches.

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Q&A• Ask questions, if any, please.

• The contents are based on chapter 16.1 of The Probabilistic Method, 3rd ed., written by Noga Alon and Joel H. Spencer.