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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 98 of 349

  • 1 - The Main Themes: Approximate Decision and Sublinear Complexity

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp 1-39

  • Summary

    Summary: In this chapter, we introduce, discuss, and illustrate the conceptual framework of property testing, emphasizing the themes of approximate decision and sublinear complexity. In particular, we discuss the key role of representation, point out the focus on properties that are not fully symmetric, present the definitions of (standard) testers and of proximity-oblivious testers (POTs), and make some general observations regarding POTs, testing, and learning. To begin, we consider the potential benefits of testing (i.e., approximate decisions of sublinear complexity).

    Section 1.1 provides a very brief introduction to property testing, sketching its basic definition and providing an overview of its flavor and potential benefits. The pace here is fast and sketchy, unlike that in the rest of this chapter. The technical material is presented in Sections 1.2 and 1.3, which constitutes the main part of this chapter. A more detailed account of the organization of this part is provided in Section 1.1.4.

    Introduction

    Big data is a broad term for data sets so large or complex that traditional data processing applications are inadequate.

    Wikipedia entry on Big Data, February 17, 2016

    Everyone talks of big data. Of course, the issue is making good use of large amounts of data, which requires analyzing it. But such an analysis may mean various things. At one extreme, it may mean locating tiny and possibly rare (but valuable) pieces of information. At the other extreme, it may means detecting global structures or estimating global parameters of the data as a whole.

    It is the latter meaning that applies to the field of property testing. This field is concerned with the analysis of global features of the data, such as determining whether the data as a whole have some global property or estimating some global parameter of their structure. The focus is on properties and parameters that go beyond simple statistics that refer to the frequency of the occurrence of various local patterns. This is not to suggest that such simple statistics are not of value, but rather that not everything of interest can be reduced to them.

    In general, the data are a set of records (or items) that may be interrelated in various ways. The contents and meaning of the data may be reflected not only in the individual items (or records), but also in the relations between them.

  • 13 - Locally Testable Codes and Proofs

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp 370-410

  • Summary

    Summary: We survey known results regarding locally testable codes and locally testable proofs (known as PCPs). Local testability refers to approximately testing large objects based on a very small number of probes, each retrieving a single bit in the representation of the object. This yields superfast approximate testing of the corresponding property (i.e., being a codeword or a valid proof).

    In terms of property testing, locally testable codes are errorcorrecting codes such that the property of being a codeword can be tested within low query complexity. As for locally testable proofs (PCPs), these can be viewed as massively parameterized properties that are testable within low query complexity such that the parameterized property is nonempty if and only if the corresponding parameter is in a predetermined set (of “valid statements”).

    Our first priority is minimizing the number of probes, and we focus on the case that this number is a constant. In this case (of a constant number of probes), we aim at minimizing the length of the constructs. That is, we seek locally testable codes and proofs of short length.

    We stress a fundamental difference between the study of locally testable codes and the study of property testing. Locally testable codes are artificially designed with the aim of making codeword testing easy. (The same holds with respect to locally testable proofs.) In contrast, property testing envisions natural objects and properties that are prescribed by an external application.

    This chapter has been adapted from our survey [130, 132], which was intended for readers who have general background in the theory of computation but may lack familiarity with property testing.We chose to maintain this feature of the original text and keep this chapter self-contained. Hence, the property testing perspective is mentioned but is not extensively relied upon. In particular, the fact that locally testable codes correspond to a special case of property testing is not pivotal to the presentation, although it is spelled out. Viewing PCPs in terms of property testing is less natural, yet this perspective is offered too (even in the foregoing summary); but again it is not pivotal to the presentation.

    This chapter also differs from the other chapters in its style: It only provides overviews of results and proofs, rather than detailed proofs. Furthermore, the footnotes provide additional details that may be more essential than in other chapters.

  • 10 - Testing Graph Properties in the General Graph Model

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp 271-303

  • 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.

  • Organization and Chapter Summaries

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp xv-xxii

  • Summary

    All chapters rely on a few core notions that are introduced in Sections 1.3.1 and 1.3.3. Although these parts of Section 1.3 provide a sufficient basis for reading any of the subsequent chapters, we strongly recommend reading the entire first chapter before proceeding to any other one.

    In contrast to the central role of Chapter 1, no other chapter is essential for reading the other chapters. In particular, interdependencies between the other chapters are rare and never heavily relied on. The main dependencies are depicted in Figure 1, where thematic dependencies are marked by solid lines and technical dependencies by dashed lines.

    Although the ordering of the chapters that follow Chapter 1 is not very important, a choice had to be made. We chose to start with simple properties of functions such as group homomorphism, low-degree polynomials, monotonicity (with respect to various partial orders), and depending on few variables (i.e., juntas). In all these cases, the correspondence between the object and its representation is transparent: the function is the object. In contrast, when moving to graph properties, the question of representation arises in an acute manner, and three different chapters are devoted to three different representations that correspond to three different testing models. Hence, from the perspective of property testing per se, it seems to make sense to start with functions and then move to graphs.

    In accordance with the foregoing, the first cluster of chapters (Chapters 2–6) deals with testing properties of functions, whereas a second cluster (Chapters 8–10) deals with testing properties of graphs. A chapter on lower bound techniques (i.e., Chapter 7) is located in between these two clusters, since lower bounds are hardly mentioned in the first cluster, whereas they appear quite prominently in Chapters 9 and 10. The reason for this phenomenon is that these lower bounds are used to justify the significantly higher complexity of some testers that are presented in Chapters 9 and 10. Indeed, in the context of this book, we view lower bounds mainly as a justification for algorithms that may be considered to have a higher than expected complexity; the lower bounds assert that this impression is actually false, and that one cannot do significantly better.

  • 8 - Testing Graph Properties in the Dense Graph Model

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp 162-212

  • Summary

    Summary: Following a general introduction to testing graph properties, this chapter focuses on the dense graph model, where graphs are represented by their adjacency matrix (predicate). The highlights of this chapter include:

    1. A presentation of a natural class of graph properties that can each be tested within query complexity that is polynomial in the reciprocal of the proximity parameter. This class, called general graph partition problems, contains properties such as t-Colorability (for any t ≥ 2) and properties that refer to the density of the max-clique and to the density of the max-cut in a graph.

    2. An exposition of the connection of testing (in this model) to Szemeredi's Regularity Lemma. The starting point and pivot of this exposition is the existence of constant-query (one-sided error) proximity-oblivious testers for all subgraph freeness properties.

    We conclude this chapter with a taxonomy of known testers, organized according to their query complexity.

    The current chapter is based on many sources; see Section 8.6.1 for details.

    Organization. The current chapter is the first of a series of three chapters that cover three models for testing graph properties. In each model, we spell out the definition of property testing (when specialized to that model), present some of the known results, and demonstrate some of the ideas involved in the construction of testers (by focusing on testing Bipartiteness, which seems a good benchmark).

    We start the current chapter with a general introduction to testing graph properties, which includes an overview of the three models (see Section 8.1.2). We then present and illustrate the “dense graph model” (Section 8.2), which is the focus of the current chapter. The main two sections (i.e., Sections 8.3 and 8.4) cover the two topics that are mentioned in the foregoing summary: Section 8.3 deals with testing arbitrary graph partition properties, as illustrated by the example of testing Bipartitness. Section 8.4 deals with the connection between property testing in this model and Szemeredi's Regularity Lemma, as illustrated by testing subgraph-freeness. The last two sections (i.e., Sections 8.5 and 8.6) are descriptive in nature: Section 8.5 presents a taxonomy of the known results, whereas Section 8.6 presents final comments.

  • Appendix C - An Index of Specific Results

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp 427-428

  • Summary

    Summary: This appendix provides an index to all results regarding specific properties that were presented in this book. For each property, we provide references only to the sections (or statements) in which relevant results can be found.

    The properties are partitioned into five main groups, whereas in each group the listing is by alphabetic order. The first list contains all properties of objects that are most naturally described as functions or sequences. The next three lists refer to the three models of testing graph properties, which were studied in Chapters 8–10. The last list refers to properties of distributions, which were studied in Chapter 11.

    Properties of Functions. Such properties were studied mostly in Chapters 2–6.

    affine functions: Last paragraph of Chapter 2.

    affine subspaces: Section 5.2.2.2 as well as Exercises 5.9–5.11.

    codewords: Chapter 13.

  • • For the Hadamard code see also linearity.

  • • For general linear codes see Proposition 1.11.

  • • For the Long Code see also dictatorship.

  • • For the Reed–Muller code see also low-degree polynomials.

    • dictatorship: Section 5.2.

    homomorphism (a.k.a. group homomorphism): Chapter 2.

    junta: Section 5.3 and Corollary 7.20.

    linearity: Special case of homomorphism.

    low-degree polynomials: Chapter 3.

    majority: Proposition 1.1.

    monomials: Section 5.2.2 as well as Corollaries 6.4 and 6.7.

    monotonicity: Chapter 4. See also sorted.

    proofs: Chapter 13.

    sparse (low-degree) polynomials (and linear functions): Corollary 7.12.

    sorted: Proposition 1.5 and 1.8. See also monotonicity.

    Graph Properties (in the Dense Graph Model). All in Chapter 8.

    biclique: Proposition 8.6.

    bipartiteness: Section 8.3.1.

    colorability (by fixed number of colors): Theorem 8.13.

    degree regularity: Theorem 8.5.

    induced subgraph freeness: Theorem 8.20.

    max-clique: Special case of Theorem 8.12.

    max-cut: Special case of Theorem 8.12.

    min-bisection: Special case of Theorem 8.12.

    subgraph freeness: Section 8.4.2.

    Recall that properties of sparse graphs (e.g., planarity) and properties that are close to any graph (e.g., connectivity) are easy to test in this model.

    Graph Properties (in the Bounded-Degree Graph Model). All in Chapter 9.

    bipartiteness: Sections 9.3.1 and 9.4.1.

    colorability (by three colors): Theorem 9.19.

    connectivity: Section 9.2.3. See Section 9.2.4 for t-connectivity.

    cycle-freeness: Section 9.2.5, Theorem 9.17, and Section 9.4.2.

    degree regularity: Section 9.2.2.

    Eulerian: See the last paragraph in Section 9.2.2 and the last paragraph in Section 9.2.3.

    expansion: Theorem 9.18.

    planarity: Special case of Theorem 9.25.

    subgraph freeness: Theorem 9.4.

Page 98 of 349