Approximating average parameters of graphs

Goldreich, Oded; Ron, Dana

doi:10.1002/rsa.20203

Cited by 55 publications

(61 citation statements)

References 18 publications

(23 reference statements)

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“…Furthermore, it seems that designing testers in this model requires the development of algorithmic techniques that may be applicable also in other areas of algorithmic research. As an example, we mention that techniques in [47] underly the average degree approximation of [38]. (Likewise techniques of [35] underly the minimum spanning tree weight approximation of [23]; indeed, as noted next, the bounded-degree incidence list model is also more algorithmic oriented than the adjacency matrix model.…”

Section: Reflectionsmentioning

confidence: 99%

Introduction to Testing Graph Properties

Goldreich

2010

Property Testing

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Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this study, and specific results are often mentioned merely as illustrations of general themes.Keywords: Graph Properties, randomized algorithms, approximation problems.This survey was originally written for inclusion in [32]. The General ContextIn general, property testing is concerned with super-fast (probabilistic) algorithms for deciding whether a given object has a predetermined property or is far from any object having this property. Such algorithms, called testers, obtain local views of the object by making adequate queries; that is, the object is seen as a function and the tester gets oracle access to this function, and thus may be expected to work in time that is sub-linear in the size of the object.Looking at the foregoing formulation, we first note that property testing is concerned with promise problems (cf. [26,30]), rather than with standard decision problems. Specifically, objects that neither have the property nor are far from having the property are discarded. The exact formulation of these promise problems refers to a distance measure defined on the set of all relevant objects (i.e., this distance measure coupled with a distance parameter determine the set of objects that are far from the property). Thus, the choice of natural distance measures is crucial to the study of property testing. Secondly, we note that the requirement that the algorithms operate in sub-linear time (i.e., without reading their entire input) calls for a specification of the type of queries that these algorithms can make to their input. Thus, the choice of natural query types is also crucial to the study of property testing. These two general considerations will become concrete once we delve into the actual subject matter (i.e., testing graph properties). Why Graphs?Let us start with an empirical observation, taken from Shimon Even's book Graph Algorithms [25] (published in 1979): 142Graph theory has long become recognized as one of the more useful mathematical subjects for the computer science student to master. The approach which is natural in computer science is the algorithmic one; our interest is not so much in existence proofs or enumeration techniques, as it is in finding efficient algorithms for solving relevant problems, or alternatively showing evidence that no such algorithms exist. Although algorithmic graph theory was started by Euler, if not earlier, its development in the last ten years has been dramatic and revolutionary.Meditating on these facts, one may ask what is the source of this ubiquitous use of graphs in computer science. The most common answer is that graphs arise naturally as a model (or an abstraction) of numerous natural and artificial objects. Another answer is that graphs help visualize binary relations over finite sets. These two different answers correspond to two types of models...

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Section: Reflectionsmentioning

confidence: 99%

Introduction to Testing Graph Properties

Goldreich

2010

Property Testing

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“…Interestingly, if one can also use neighborhood queries, then it is possible to approximate the average degree using O( √ n/ε O(1) ) queries with a ratio of (1 + ε), as shown by Goldreich and Ron [37]. The model for neighborhood queries is as follows.…”

Section: Approximating the Average Degreementioning

confidence: 99%

“…We remark that one can simulate degree queries in this model with O(log n) queries. Therefore, the algorithm from [37] uses only neighbor queries.…”

Section: Approximating the Average Degreementioning

confidence: 99%

“…By our analysis above, we know that there are only few edges within H and that we make only a small error in estimating the number of edges within L. We loose the factor of two, because we "see" edges from L to H only from one side. The idea behind the algorithm from [37] is to approximate the fraction of edges from L to H and add it to the final estimate. This has the effect that we count any edge between L and H twice, canceling the effect that we see it only from one side.…”

Section: Approximating the Average Degreementioning

confidence: 99%

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Sublinear-time Algorithms

Czumaj

Sohler

2010

Property Testing

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“…We assume for simplicity that the algorithms are given d as input. However, they actually need only a constant factor estimate of d, and this can be obtained by performingÕ( n/d) degree queries in expectation [12,17] (which is negligble for our algorithms). Figure 1.1: A schematic illustration of the query complexity for the different query types (and one-sided error) on a "log-log" scale.…”

Section: Related Work Onmentioning

confidence: 99%

Comparing the Strength of Query Types in Property Testing: The Case of Testing k-Colorability

et al. 2010

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Abstract. We study the power of four query models in the context of property testing in general graphs, where our main case study is the problem of testing k-colorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the i th neighbor of a particular vertex. We show that while for pair queries, testing kcolorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give upper and lower bounds on the query complexity for one-sided error in all the models, where the bounds are nearly tight for three of the models. In some of the cases our lower bounds extend to two-sided error algorithms. The problem of testing k-colorability was previously studied in the contexts of dense graphs and of sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.

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Approximating average parameters of graphs

Cited by 55 publications

References 18 publications

Introduction to Testing Graph Properties

Introduction to Testing Graph Properties

Sublinear-time Algorithms

Comparing the Strength of Query Types in Property Testing: The Case of Testing k-Colorability

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