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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

    Advisor: Professor Maw-Shang Chang

    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.

    A branch of Ramsey theory (see also Van der Waerdenstheorem).

    How about N(3, 1/2)?

    {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.

    A branch of Ramsey theory (see also Van der Waerdenstheorem).

    How about N(3, 1/2)?

    {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.

    A branch of Ramsey theory (see also Van der Waerdenstheorem).

    How about N(3, 1/2)?

    {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.

    A branch of Ramsey theory (see also Van der Waerdenstheorem).

    How about N(3, 1/2)?

    {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

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