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23
Algorithmic and Analysis Techniques in Property Testing
"... Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform ..."
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Cited by 27 (4 self)
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Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this article we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: • The selfcorrecting approach, which was mainly applied in the study of property testing of algebraic properties; • The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the densegraphs model), as well as in other contexts;
Sublinear time algorithms
 SIGACT News
, 2003
"... Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algo ..."
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Cited by 25 (3 self)
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Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algorithmic research is to design efficient algorithms, where efficiency is typicallymeasured as a function of the length of the input. For instance, the elementary school algorithm for multiplying two n digit integers takes roughly n2 steps, while more sophisticated algorithmshave been devised which run in less than n log2 n steps. It is still not known whether a linear time algorithm is achievable for integer multiplication. Obviously any algorithm for this task, as for anyother nontrivial task, would need to take at least linear time in n, since this is what it would take to read the entire input and write the output. Thus, showing the existence of a linear time algorithmfor a problem was traditionally considered to be the gold standard of achievement. Nevertheless, due to the recent tremendous increase in computational power that is inundatingus with a multitude of data, we are now encountering a paradigm shift from traditional computational models. The scale of these data sets, coupled with the typical situation in which there is verylittle time to perform our computations, raises the issue of whether there is time to consider any more than a miniscule fraction of the data in our computations? Analogous to the reasoning thatwe used for multiplication, for most natural problems, an algorithm which runs in sublinear time must necessarily use randomization and must give an answer which is in some sense imprecise.Nevertheless, there are many situations in which a fast approximate solution is more useful than a slower exact solution.
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
"... ..."
Sublineartime algorithms
 In Oded Goldreich, editor, Property Testing, volume 6390 of Lecture Notes in Computer Science
, 2010
"... In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1 ..."
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Cited by 12 (3 self)
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In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1
On Proximity Oblivious Testing
, 2009
"... We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameter, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximityobli ..."
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Cited by 9 (3 self)
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We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameter, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximityoblivious testers. While proximityoblivious testers were studied before – most notably in the algebraic setting – the current study seems to be the first one to focus on graph properties. We provide a mix of positive and negative results, and in particular characterizations of the graph properties that have constantquery proximityoblivious testers in the two standard models (i.e., the adjacency matrix and the boundeddegree models). Furthermore, we show that constantquery proximityoblivious testers do not exist for many easily testable properties, and that even when proximityoblivious testers exist, repeating them does not necessarily yield the best standard testers for the corresponding property.
Introduction to testing graph properties
 In Property Testing
, 2010
"... 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 ..."
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Cited by 8 (0 self)
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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
Local Graph Partitions for Approximation and Testing
"... Abstract—We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of boundeddegree graphs with an excluded minor, and in general, for any hyperfinite class of boundeddegree graphs. These oracles utilize onl ..."
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Cited by 7 (1 self)
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Abstract—We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of boundeddegree graphs with an excluded minor, and in general, for any hyperfinite class of boundeddegree graphs. These oracles utilize only local computation to consistently answer queries about a global partition that breaks the graph into small connected components by removing only a small fraction of the edges. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance: • We give constanttime approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor. • We show a simple proof that any minorclosed graph property is testable in constant time in the bounded degree model. • We prove that it is possible to approximate the distance to almost any hereditary property in any bounded degree hereditary families of graphs. Hereditary properties of interest include bipartiteness, kcolorability, and perfectness. 1.
Local Monotonicity Reconstruction
"... We investigate the problem of monotonicity reconstruction, as defined by Ailon, Chazelle, Comandur and Liu (2004) in a localized setting. We have oracle access to a nonnegative realvalued function f defined on the domain [n] d = {1,..., n} d (where d is viewed as a constant). We would like to closel ..."
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Cited by 7 (0 self)
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We investigate the problem of monotonicity reconstruction, as defined by Ailon, Chazelle, Comandur and Liu (2004) in a localized setting. We have oracle access to a nonnegative realvalued function f defined on the domain [n] d = {1,..., n} d (where d is viewed as a constant). We would like to closely approximate f by a monotone function g. This should be done by a procedure (a filter) that given as input a point x ∈ [n] d outputs the value of g(x), and runs in time that is polylogarithmic in n. The procedure can (indeed must) be randomized, but we require that all of the randomness be specified in advance by a single short random seed. We construct such an implementation where the time and space per query is (log n) O(1) and the size of the seed is polynomial in log n and d. Furthermore, with high probability, the ratio of the (Hamming) distance between g and f to the minimum possible Hamming distance between a monotone function and f is bounded above by a function of d (independent of n). This allows for a local implementation: one can initialize many copies of the filter with the same short random seed, and they can autonomously handle queries, while producing outputs that are consistent with the same approximating function g.
Hierarchy Theorems for Property Testing
, 2009
"... Referring to the query complexity of property testing, we prove the existence of a rich hierarchy of corresponding complexity classes. That is, for any relevant function q, we prove the existence of properties that have testing complexity Θ(q). Such results are proven in three standard domains often ..."
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Cited by 5 (3 self)
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Referring to the query complexity of property testing, we prove the existence of a rich hierarchy of corresponding complexity classes. That is, for any relevant function q, we prove the existence of properties that have testing complexity Θ(q). Such results are proven in three standard domains often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing boundeddegree graphs. While in two cases the proofs are quite straightforward, the techniques employed in the case of the dense graph model seem significantly more involved. Specifically, problems that arise and are treated in the latter case include (1) the preservation of distances between graph under a blowup operation, and (2) the
An Analytic Approach to Stability
, 2010
"... The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order n can be made isomorphic by changing o(n2) edges. Here we show h ..."
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Cited by 5 (0 self)
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The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order n can be made isomorphic by changing o(n2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erdős–Simonovits Stability Theorem. Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fractional versions are within a constant factor from each other, thus answering a question of Goldreich, Krivelevich, Newman, and Rozenberg. 1