Introduction To Property Testing 1st Edition by Oded Goldreich ISBN 1107194059 9781107194052

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ISBN 10: 1107194059 
ISBN 13: 9781107194052
Author: Oded Goldreich

Introduction To Property Testing 1st Table of contents:

1 The Main Themes: Approximate Decision and Sublinear Complexity
1.1 Introduction
1.1.1 Property Testing at a Glance
1.1.2 On the Potential Benefits of Property Testers
1.1.3 On the Flavor of Property Testing Research
1.1.4 Organization and Some Notations
1.2 Approximate Decisions
1.2.1 A Detour: Approximate Search Problems
1.2.2 Property Testing: Approximate Decision Problems
1.2.3 Property Testing: Sublinear Complexity
1.2.4 Symmetries and Invariants
1.2.5 Objects and Representation
1.3 Notions, Definitions, Goals, and Basic Observations
1.3.1 Basics
1.3.2 Ramifications
1.3.3 Proximity-Oblivious Testers (POTs)
1.3.4 The Algebra of Property Testing
1.3.5 Testing via Learning
1.4 Historical Notes
1.5 Suggested Reading and Exercises
Basic Exercises
Additional Exercises
1.6 Digest: The Most Important Points
2 Testing Linearity (Group Homomorphism)
2.1 Preliminaries
2.2 The Tester
2.3 Chapter Notes
3 Low-Degree Tests
3.1 A Brief Introduction
3.2 A Kind of Intuition (which may be skipped)
3.2.1 The Univariate Case
3.2.2 The Multivariate Case
3.2.3 Linking the Above Intuition to the Actual Proof
3.3 Background
3.4 The Tester
3.4.1 Analysis of the Tester
3.4.2 Digest (or an Abstraction)
3.5 Chapter Notes
Exercises
4 Testing Monotonicity
4.1 Introduction
4.2 Boolean Functions on the Boolean Hypercube
4.2.1 The Edge Test
4.2.2 Path Tests
4.3 Multivalued Functions on the Discrete Line
4.3.1 A Tester Based on Binary Search
4.3.2 Other Testers
4.4 Multivalued Functions on the Hypergrid
4.4.1 Dimension Reduction (Proof of Lemma 4.13)
4.4.2 Range Reduction (Overview of the Proof of Lemma 4.14)
4.5 Chapter Notes
4.5.1 History and Credits
4.5.2 Related Problems
4.5.3 Exercises
5 Testing Dictatorships, Juntas, and Monomials
5.1 Introduction
5.2 Testing Dictatorship via Self-correction
5.2.1 The Tester
5.2.2 Testing Monomials
5.2.3 The Self-correction Paradigm: An Abstraction
5.3 Testing Juntas
5.4 Chapter Notes
Basic Exercises
Additional Exercises
6 Testing by Implicit Sampling
6.1 Introduction
6.2 Testing Subsets of k-Juntas
6.3 Extension to Properties Approximated by Subsets of k-Juntas
6.4 Chapter Notes
On Testing Problems Associated with Sets of Boolean Functions
Exercises
7 Lower Bounds Techniques
7.1 Introduction
7.2 Indistinguishability of Distributions
7.2.1 The Actual Method
7.2.2 Illustrating the Application of the Method
7.2.3 Further Reflections
7.3 Reduction from Communication Complexity
7.3.1 Communication Complexity
7.3.2 The Methodology
7.3.3 Illustrating the Application of the Methodology
7.4 Reduction among Testing Problems
7.5 Lower Bounds for Restricted Testers
7.5.1 One-Sided Error Testers
7.5.2 Nonadaptive Testers
7.6 Chapter Notes
Exercises
8 Testing Graph Properties in the Dense Graph Model
8.1 The General Context: Introduction to Testing Graph Properties
8.1.1 Basic Background
8.1.2 Three Models of Testing Graph Properties
8.2 The Dense Graph Model: Some Basics
8.2.1 The Actual Definition
8.2.2 Abuses of the Model: Trivial and Sparse Properties
8.2.3 Testing Degree Regularity
8.2.4 Digest: Levin’s Economical Work Investment Strategy
8.3 Graph Partition Problems
8.3.1 Testing Bipartiteness
8.3.2 The Actual Definition and the General Result
8.4 Connection to Szemeŕedi’s Regularity Lemma
8.4.1 The Regularity Lemma
8.4.2 Subgraph Freeness
8.4.3 The Structure of Properties That Have Size-Oblivious Testers
8.5 A Taxonomy of the Known Results
8.5.1 Testability in q(ε) Queries, for any Function q
8.5.2 Testability in Poly(1/ε) Queries
8.5.3 Testability in Ṍ(1/ε) Queries
8.5.4 Additional Issues
8.6 Chapter Notes
8.6.1 Historical Perspective and Credits
8.6.2 Testing versus Other Forms of Approximation
8.6.3 A Contrast with Recognizing Graph Properties
8.6.4 Exercises
9 Testing Graph Properties in the Bounded-Degree Graph Model
9.1 The Bounded-Degree Model: Definitions and Issues
9.2 Testing by a Local Search
9.2.1 Testing Subgraph Freeness
9.2.2 Testing Degree Regularity
9.2.3 Testing Connectivity
9.2.4 Testing t-Connectivity (Overview and One Detail)
9.2.5 Testing Cycle-Freeness (with Two-Sided Error)
9.3 Lower Bounds
9.3.1 Bipartiteness
9.3.2 Applications to Other Properties
9.3.3 Linear Lower Bounds
9.4 Testing by Random Walks
9.4.1 Testing Bipartiteness
9.4.2 One-Sided Error Tester for Cycle-Freeness
9.5 Testing by Implementing and Utilizing Partition Oracles
9.5.1 The Simpler Implementation
9.5.2 The Better Implementation
9.6 A Taxonomy of the Known Results
9.6.1 Testability in q(ε) Queries, for Any Function q
9.6.2 Testability in Ṍ(k[sup(1/2)]) · poly(1/ε) Queries
9.6.3 Additional Issues
9.7 Chapter Notes
9.7.1 Historical Perspective and Credits
9.7.2 Directed Graphs
9.7.3 Exercises
10 Testing Graph Properties in the General Graph Model
10.1 The General Graph Model: Definitions and Issues
10.1.1 Perspective: Comparison to the Two Previous Models
10.1.2 The Actual Definition
10.2 On Obtaining Testers for the Current Model
10.2.1 An Explicit Adaptation: The Case of Connectivity
10.2.2 Using a Reduction: The Case of Bipartiteness
10.3 Estimating the Average Degree and Selecting Random Edges
10.3.1 Lower Bounds
10.3.2 Algorithms
10.4 Using Adjacency Queries: The Case of Bipartiteness
10.5 Chapter Notes
10.5.1 Gaps between the General Graph Model and the Bounded-Degree Model
10.5.2 History and Credits
10.5.3 Reflections
10.5.4 Exercises
11 Testing Properties of Distributions
11.1 The Model
11.1.1 Testing Properties of Single Distributions
11.1.2 Testing Properties of Pairs of Distributions
11.1.3 Label-invariant Properties
11.1.4 Organization
11.2 Testing Equality to a Fixed Distribution
11.2.1 The Collision Probability Tester and Its Analysis
11.2.2 The General Case (Treated by a Reduction to Testing Uniformity)
11.2.3 A Lower Bound
11.3 Testing Equality between Two Unknown Distributions
11.3.1 Detour: Poisson Distributions
11.3.2 The Actual Algorithm and Its Analysis
11.3.3 Applications: Reduction to the Case of Small Norms
11.4 On the Complexity of Testing Properties of Distributions
11.5 Chapter Notes
11.5.1 History and Credits
11.5.2 Exercises
12 Ramifications and Related Topics
12.1 Tolerant Testing and Distance Approximation
12.2 Additional Promises on the Input
12.3 Sample-Based Testers
12.4 Testing with Respect to Other Distance Measures
12.5 Local Computation Algorithms
12.5.1 Definitions
12.5.2 Finding Huge Structures
12.5.3 Local Reconstruction
12.6 Noninteractive Proofs of Proximity (MAPs)
12.7 Chapter Notes
12.7.1 Historical Notes
12.7.2 Massively Parameterized Properties
12.7.3 Exercises
13 Locally Testable Codes and Proofs
13.1 Introduction
13.2 Definitions
13.2.1 Codeword Testers
13.2.2 Proof Testers
13.2.3 Ramifications and Relation to Property Testing
13.2.4 On Relating Locally Testable Codes and Proofs
13.3 Results and Ideas
13.3.1 The Mere Existence of Locally Testable Codes and Proofs
13.3.2 Locally Testable Codes and Proofs of Polynomial Length
13.3.3 Locally Testable Codes and Proofs of Nearly Linear Length
13.4 Chapter Notes
13.4.1 Historical Notes
13.4.2 On Obtaining Superfast testers
13.4.3 The Alternative Regime: LTCs of Linear Length
13.4.4 Locally Decodable Codes
13.4.5 Exercises
Appendix A: Probabilistic Preliminaries
A.1 Notational Conventions
A.2 Some Basic Notions and Facts
A.3 Basic Facts Regarding Expectation and Variance
A.4 Three Inequalities
A.4.1 Markov’s Inequality
A.4.2 Chebyshev’s Inequality
A.4.3 Chernoff Bound
A.4.4 Pairwise Independent versus Totally Independent Sampling

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Tags: Oded Goldreich, Property Testing