Testing the diameter of graphs

Parnas, Michal; Ron, Dana

doi:10.1002/rsa.10013

Cited by 64 publications

(54 citation statements)

References 12 publications

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“…Turning to the specific results, we mention that the adaptation of the connectivity tester to the current model is due to [14]. The results regarding testing Bipartitenss in the general graph model were obtained by Kaufman, Krivelevich, and Ron [13].…”

Section: History and Creditsmentioning

confidence: 84%

“…The study of property testing in the general graph model was initiated by Parnas and Ron [14], who only considered incidence queries, and extended by Kaufman, Krivelevich, and Ron [13], who considered both types of queries. 30 Needless to say, the aim of these works was to address the limitations of the previous models for testing graph properties; that is, to allow the consideration of arbitrary graphs.…”

Section: History and Creditsmentioning

confidence: 99%

“…Of special interest are graphs that are neither very dense nor have a bounded degree. In contrast, research in testing properties of graphs started (in [8]) with the study of dense graphs, proceeded to the study of bounded-degree graphs (in [9]), and reached general graphs only in [14,13]. This evolution has historical reasons, which will be reviewed next.…”

Section: Reflectionsmentioning

confidence: 99%

“…The identification of graphs with a specific functional representation was abandoned by Parnas and Ron [14], who developed a more general model by decoupling the type of queries allowed to the tester from the distance measure: Whatever is the mechanism of accessing the graph, the distance between graphs is defined as the number of edges in their symmetric difference (rather than the number of different entries with respect to some specific functional representation). Furthermore, the relative distance is defined as the size of the symmetric difference divided by the actual (total) number of edges in both graphs (rather than divided by some (possibly non-tight) upper-bound on the latter quantity).…”

Section: Reflectionsmentioning

confidence: 99%

“…To summarize, we advocate further study of the model of [14,13] for two reasons. The first reason is that we believe in the greater relevance of this model to computer science applications.…”

Section: Reflectionsmentioning

confidence: 99%

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Testing Graph Properties in the Dense Graph Model

Goldreich¹

Introduction to Property Testing

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This lecture 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 lecture include:Teaching note: Again, we recommend covering only part of the material in class, and leaving the rest for optional independent reading. Aside from Section 1, which seems a must, the choice of what to teach and what to leave out is less clear. If pressed for our own choice, then the fact is that we chose to cover Sections 2.2 and 3.2.1. 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 last two lectures (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.Giving up on the representation as a yardstick for the relative distance between graphs, leaves us with no absolute point of reference. Instead, we just define the relative distance between graphs in relation to the actual number of edges in these graphs; specifically, the relative distance between the graphs G = ([k], E) and G ′ = ([k], E ′ ) may be defined asdenotes the symmetric difference between E and E ′ . Indeed, the normalization by max(|E|, |E ′ |) is somewhat arbitrary, and alternatives that seem as natural include |E| + |E ′ |, |E ∪ E ′ | and (|E| + |E ′ |)/2; yet, all these alternatives are within a factor of two from one another (and the slackness is even smaller in the typical case where |E ∪ E ′ | ≈ |E ∩ E ′ |).Turning to the question of query types, we again need to make a choice, which is now free from representational considerations. The most natural choice is to allow both incidence queries and adjacency queries; that is, we allow the two types of queries that were each allowed in one of the two previous models. Hence, the graph G = ([k], E) is (redundantly) represented by (or rather accessed via) two functions:1. An incidence function g 1 : [k] × [k − 1] → {0, 1, ..., k} such that g 1 (u, i) = 0 if u has less than i neighbors and g 1 (Indeed, here k − 1 serves as a (trivial) degree bound. (Recall that the bounded-degree graph model relied on an explicit degree bound, which was denoted d.)2. An adjacency predicate g 2 : [k] × [k] → {0, 1} such that g 2 (u, v) = 1 if and only if {u, v} ∈ E.Typically, adjacency queries become more useful when the graph becomes more dense, whereas incidence queries (a.k.a neighbor queries) become more useful when the graph becomes more sparse (c...

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Section: History and Creditsmentioning

confidence: 84%

Section: History and Creditsmentioning

confidence: 99%

Section: Reflectionsmentioning

confidence: 99%

Section: Reflectionsmentioning

confidence: 99%

Section: Reflectionsmentioning

confidence: 99%

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Testing Graph Properties in the Dense Graph Model

Goldreich¹

Introduction to Property Testing

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Testing membership in parenthesis languages

Parnas¹,

Ron²,

Rubinfeld³

2002

Random Struct Algorithms

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ABSTRACT:We continue the investigation of properties defined by formal languages. This study was initiated by Alon et al.[1], who described an algorithm for testing properties defined by regular languages. Alon et al. also considered several context free languages, and in particular Dyck languages, which contain strings of properly balanced parentheses. They showed that the first Dyck language, which contains strings over a single type of pairs of parentheses, is testable in time independent of n, where n is the length of the input string. However, the second Dyck language, defined over two types of parentheses, requires ⍀ (log n) queries. Here we describe a sublinear-time algorithm for testing all Dyck languages. Specifically, the running time of our algorithm is Õ (n 2/3 /⑀ 3 ), where ⑀ is the given distance parameter. Furthermore, we improve the lower bound for testing Dyck languages to⍀(n 1/11 ) for constant ⑀. We also describe a testing algorithm for the context free language L REV ϭ {uu r vv r : u, v ʦ ⌺*}, where ⌺ is a fixed alphabet. The running time of our algorithm is Õ ( ͌ n/⑀), which almost matches the lower bound given by Alon et al. [1].

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Approximating average parameters of graphs

Goldreich

Ron

2008

Random Struct Algorithms

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Inspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph. Specifically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph. Since our focus is on sublinear algorithms, these algorithms access the input graph via queries to an adequate oracle.We consider two types of queries. The first type is standard neighborhood queries (i.e., what is the ith neighbor of vertex v?), whereas the second type are queries regarding the quantities that we need to find the average of (i.e., what is the degree of vertex v? and what is the distance between u and v?, respectively).Loosely speaking, our results indicate a difference between the two problems: For approximating the average degree, the standard neighbor queries suffice and in fact are preferable to degree queries. In contrast, for approximating average distances, the standard neighbor queries are of little help whereas distance queries are crucial.

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Testing the diameter of graphs

Cited by 64 publications

References 12 publications

Testing Graph Properties in the Dense Graph Model

Testing Graph Properties in the Dense Graph Model

Testing membership in parenthesis languages

Approximating average parameters of graphs

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