regularity lemma

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Its about Regularity Lemma in Randomized Algorithms

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• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction to the Regularity Lemma

Speaker: Joseph, Chuang-Chieh Lin

Computation Theory LaboratoryDept. Computer Science and Information Engineering

National Chung Cheng University, Taiwan

July 8, 2008

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredis Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredis Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredis Theorom)

Let k be a positive integer and let 0 < < 1. Then thereexists a positive integer N = N(k, ), such that for everyA {1, 2, ...,N}, |A| N, A contains an arithmeticprogression of length k.

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredis Theorom)

Let k be a positive integer and let 0 < < 1. Then thereexists a positive integer N = N(k, ), such that for everyA {1, 2, ...,N}, |A| N, A contains an arithmeticprogression of length k.

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredis Theorom)

Let k be a positive integer and let 0 < < 1. Then thereexists a positive integer N = N(k, ), such that for everyA {1, 2, ...,N}, |A| N, A contains an arithmeticprogression of length k.

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredis Theorom)

Let k be a positive integer and let 0 < < 1. Then thereexists a positive integer N = N(k, ), such that for everyA {1, 2, ...,N}, |A| N, A contains an arithmeticprogression of length k.

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Related to a famous result...

Theorem

The primes contain arbitrarily long

arithmetic progressions.

(Terence Tao and Ben J. Green, 2004)

Terence Tao (2006 Fields Medal)

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, ):

C log(1/)k1

N(k , ) 2222

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredis Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, ):

C log(1/)k1

N(k , ) 2222

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredis Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, ):

C log(1/)k1

N(k , ) 2222

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredis Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, ):

C log(1/)k1

N(k , ) 2222

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredis Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

Endre Szemerei

About 15 years later, its power was noted and plenty results ingraph theory and theoretical computer science have beenworked out.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredis Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Density of bipartite graphs

Definition 2.1

Given a bipartite graph G = (A,B ,E ), E A B . Thedensity of G is defined to be

d(A,B) =e(A,B)

|A| |B |,

where e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

d(A,B) = 1 If G is complete bipartite.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

Density of bipartite graphs

Definition 2.1

Given a bipartite graph G = (A,B ,E ), E A B . Thedensity of G is defined to be

d(A,B) =e(A,B)

|A| |B |,

where e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

d(A,B) = 1 If G is complete bipartite.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

-regular pair

Definition 2.2

Let > 0. Given a graph G and two disjoint vertex sets A V ,B V , we say that the pair (A,B) is -regular if for every X Aand Y B satisfying

|X | |A| and |Y | |B |,

we have|d(X ,Y ) d(A,B)| < .

If G = (A,B ,E ) is a complete bipartite graph, then (A,B) is-regular for every > 0.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• IntroductionRegular pairs and their properties

Szemeredis Regularity LemmaA simple application

Conclusion and remarks

-regular pair (contd.)

1/2-regular 1/2-irregular 1/2-irregular

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

• Intro