Book contents
- Frontmatter
- Contents
- Preface
- Organization and Chapter Summaries
- Notation
- Acknowledgments
- 1 The Main Themes: Approximate Decision and Sublinear Complexity
- 2 Testing Linearity (Group Homomorphism)
- 3 Low-Degree Tests
- 4 Testing Monotonicity
- 5 Testing Dictatorships, Juntas, and Monomials
- 6 Testing by Implicit Sampling
- 7 Lower Bounds Techniques
- 8 Testing Graph Properties in the Dense Graph Model
- 9 Testing Graph Properties in the Bounded-Degree Graph Model
- 10 Testing Graph Properties in the General Graph Model
- 11 Testing Properties of Distributions
- 12 Ramifications and Related Topics
- 13 Locally Testable Codes and Proofs
- Appendix A Probabilistic Preliminaries
- Appendix B A Mini-Compendium of General Results
- Appendix C An Index of Specific Results
- References
- Index
10 - Testing Graph Properties in the General Graph Model
Published online by Cambridge University Press: 13 November 2017
- Frontmatter
- Contents
- Preface
- Organization and Chapter Summaries
- Notation
- Acknowledgments
- 1 The Main Themes: Approximate Decision and Sublinear Complexity
- 2 Testing Linearity (Group Homomorphism)
- 3 Low-Degree Tests
- 4 Testing Monotonicity
- 5 Testing Dictatorships, Juntas, and Monomials
- 6 Testing by Implicit Sampling
- 7 Lower Bounds Techniques
- 8 Testing Graph Properties in the Dense Graph Model
- 9 Testing Graph Properties in the Bounded-Degree Graph Model
- 10 Testing Graph Properties in the General Graph Model
- 11 Testing Properties of Distributions
- 12 Ramifications and Related Topics
- 13 Locally Testable Codes and Proofs
- Appendix A Probabilistic Preliminaries
- Appendix B A Mini-Compendium of General Results
- Appendix C An Index of Specific Results
- References
- Index
Summary
Summary: This chapter is devoted to testing graph properties in the general graph model, where graphs are inspected via incidence and adjacency queries, and distances between graphs are normalized by their actual size (i.e., actual number of edges). The highlights of this chapter include
1. Demonstrating the derivation of testers for this model from testers for the bounded-degree graph model
2. Studying the tasks of estimating the number of edges in a graph and sampling edges uniformly at random
We conclude this chapter with some reflections regarding the three models of testing graph properties.
The current chapter is based on several sources; see Section 10.5.2 for details.
Organization. Following an introduction to the general graph model (Section 10.1), we study the issues that arise when trying to extend testers for the bounded-degree graph model to testers for the current model (Section 10.2). Next, in Section 10.3, we study the related problems of estimating the average degree in a general graph and selecting random edges in it, presenting two different algorithmic approaches toward solving these problems (see Sections 10.3.2.1 and 10.3.2.2, respectively). As illustrated in Section 10.2.2, these problems are pivotal for the design of some testers. Lastly, in Section 10.4, we illustrate the possible benefits of using both incidence and gadjacency queries.
The General Graph Model: Definitions and Issues
The general graph model is intended to capture arbitrary graphs, which may be neither dense nor of bounded degree. Such graphs occur most naturally in many settings, but they are not captured (or not captured well) by the models presented in the previous two chapters (i.e., the dense graph model and the bounded-degree graph model).
Recall that both in the dense graph model and in the bounded-degree graph model, the query types (i.e., ways of probing the tested graph) and the distance measure (i.e., distance between graphs) were linked to the representation of graphs as functions. In contrast to these two models, in the general graph model the representation is blurred, and the query types and distance measure are decoupled.
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- Introduction to Property Testing , pp. 271 - 303Publisher: Cambridge University PressPrint publication year: 2017