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20
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 48 (7 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;
Every minorclosed property of sparse graphs is testable
, 2007
"... Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a sim ..."
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Cited by 36 (3 self)
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Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minorclosed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outerplanar, seriesparallel, bounded genus, bounded treewidth and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1
Testing hereditary properties of nonexpanding boundeddegree graphs
"... We study graph properties which are testable for bounded degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is believed that almost all, even very simpl ..."
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Cited by 22 (8 self)
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We study graph properties which are testable for bounded degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is believed that almost all, even very simple graph properties require a large complexity to be tested for arbitrary (bounded degree) graphs. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. We call a graph family nonexpanding if every graph in this family has a weak expansion (its expansion is O(1 / log 2 n), where n is the graph size). A graph family is hereditary if it is closed under vertex removal. Similarly, a graph property is hereditary if it is closed under vertex removal. We call a graph property Π to be testable for a graph family F if for every graph G ∈ F, in time independent of the size of G we can distinguish between the case when G satisfies property Π and when it is far from every graph satisfying property Π. In this paper we prove that in the bounded degree graph model, any hereditary property is testable if the input graph belongs to a hereditary and nonexpanding family of graphs. As an application, our result implies that, for example, any hereditary property (e.g., kcolorability,
A proof of Green’s conjecture regarding the removal properties of sets of linear equations
"... A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homo ..."
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Cited by 20 (1 self)
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A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green’s conjecture by showing that every set of linear equations (even nonhomogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [7] related to algorithms for testing properties of boolean functions.
Testing expansion in boundeddegree graphs
 Proc. of FOCS 2007
"... We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertexexpansion: an αexpander is a graph G = (V, E) in which every subset U ⊆ V of at most V /2 vertices has a neighborhood of size at least α · U. Our main result is that one can distinguish good ..."
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Cited by 16 (1 self)
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We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertexexpansion: an αexpander is a graph G = (V, E) in which every subset U ⊆ V of at most V /2 vertices has a neighborhood of size at least α · U. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time Õ(√n). We prove that the property testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every αexpander with probability at least 2, where α ∗ ( α = Θ 2 d2) and d is the maximum degree of the graphs. The algorithm assumes the log(n/ε) boundeddegree) graphs model with adjacency list graph representation and its running time is O and rejects every graph that is εfar from any α∗expander with probability at least 2 3 d 2 √ n log(n/ε) α 2 ε 3
Testing the expansion of a graph
, 2007
"... We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an αexpander if every vertex set U ⊂ V of size at most 1V  has a neigh2 borhood of size at least αU. We show that the algorithm proposed by Goldreich and Ron [9] (ECCC2000) f ..."
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Cited by 10 (1 self)
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We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an αexpander if every vertex set U ⊂ V of size at most 1V  has a neigh2 borhood of size at least αU. We show that the algorithm proposed by Goldreich and Ron [9] (ECCC2000) for testing the expansion of a graph distinguishes with high probability between αexpanders of degree bound d and graphs which are ɛfar from having expansion at least Ω(α2). This improves a recent result of Czumaj and Sohler [3] (FOCS07) who showed that this algorithm can distinguish between αexpanders of degree bound d and graphs which are ɛfar from having expansion at least Ω(α2 / log n). It also improves a recent result of Kale and Seshadhri [12] (ECCC2007) who showed that this algorithm can distinguish between αexpanders and graphs which are ɛfar from having expansion at least Ω(α2) with twice the maximum degree. Our methods combine the techniques of [3], [9] and [12]. 1
Making queries tractable on big data with preprocessing
 In VLDB
, 2013
"... A query class is traditionally considered tractable if there exists a polynomialtime (PTIME) algorithm to answer its queries. When it comes to big data, however, PTIME algorithms often become infeasible in practice. A traditional and effective approach to coping with this is to preprocess data off ..."
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Cited by 6 (3 self)
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A query class is traditionally considered tractable if there exists a polynomialtime (PTIME) algorithm to answer its queries. When it comes to big data, however, PTIME algorithms often become infeasible in practice. A traditional and effective approach to coping with this is to preprocess data offline, so that queries in the class can be subsequently evaluated on the data efficiently. This paper aims to provide a formal foundation for this approach in terms of computational complexity. (1) We propose a set of Πtractable queries, denoted by ΠT0Q, to characterize classes of queries that can be answered in parallel polylogarithmic time (NC) after PTIME preprocessing. (2) We show that several natural query classes are Πtractable and are feasible on big data. (3) We also study a set ΠTQ of query classes that can be effectively converted to Πtractable queries by refactorizing its data and queries for preprocessing. We introduce a form of NC reductions to characterize such conversions. (4) We show that a natural query class is complete for ΠTQ. (5) We also show that ΠT0Q ⊂ P unless P = NC, i.e., the set ΠT0Q of all Πtractable queries is properly contained in the set P of all PTIME queries. Nonetheless, ΠTQ = P, i.e., all PTIME query classes can be made Πtractable via proper refactorizations. This work is a step towards understanding the tractability of queries in the context of big data. 1.
LOWER BOUNDS FOR MULTIPASS PROCESSING OF MULTIPLE DATA STREAMS
 SUBMITTED TO THE SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
"... This paper gives a brief overview of computation models for data stream processing, and it introduces a new model for multipass processing of multiple streams, the socalled mp2sautomata. Two algorithms for solving the set disjointness problem with these automata are presented. The main technica ..."
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This paper gives a brief overview of computation models for data stream processing, and it introduces a new model for multipass processing of multiple streams, the socalled mp2sautomata. Two algorithms for solving the set disjointness problem with these automata are presented. The main technical contribution of this paper is the proof of a lower bound on the size of memory and the number of heads that are required for solving the set disjointness problem with mp2sautomata.
Robust characterizations of kwise independence over product spaces and related testing results. Random Structures and Algorithms
, 2012
"... A discrete distribution D over Σ1 × · · · × Σn is called (nonuniform) kwise independent if for any subset of k indices {i1,..., ik} and for any z1 ∈ Σi1,..., zk ∈ Σik, PrX∼D[Xi1 · · ·Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · ·PrX∼D[Xik = zk]. We study the problem of testing (nonuniform) k ..."
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Cited by 1 (1 self)
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A discrete distribution D over Σ1 × · · · × Σn is called (nonuniform) kwise independent if for any subset of k indices {i1,..., ik} and for any z1 ∈ Σi1,..., zk ∈ Σik, PrX∼D[Xi1 · · ·Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · ·PrX∼D[Xik = zk]. We study the problem of testing (nonuniform) kwise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from kwise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0, 1}n. For the nonuniform case, we give a new characterization of distributions being kwise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. [STOC’07, pp. 496–505] on uniform kwise independence over the Boolean cubes to nonuniform kwise independence over product spaces. Our results yield natural testing algorithms for kwise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.
Testing Nonuniform kwise Independent Distributions over Product Spaces
"... A discrete distribution D over Σ1 × · · · × Σn is called (nonuniform) kwise independent if for any set of k indexes {i1,..., ik} and for any z1 ∈ Σi1,..., zk ∈ Σik, PrX∼D[Xi1 · · · Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · · PrX∼D[Xik = zk]. We study the problem of testing (nonuniform) ..."
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Cited by 1 (1 self)
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A discrete distribution D over Σ1 × · · · × Σn is called (nonuniform) kwise independent if for any set of k indexes {i1,..., ik} and for any z1 ∈ Σi1,..., zk ∈ Σik, PrX∼D[Xi1 · · · Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · · PrX∼D[Xik = zk]. We study the problem of testing (nonuniform) kwise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from kwise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0, 1} n. For the nonuniform case, we give a new characterization of distributions being kwise independent and further show that such a characterization is robust based on our results for the uniform case. These greatly generalize the results of Alon et al. [1] on uniform kwise independence over the Boolean cubes to nonuniform kwise independence over product spaces. Our results yield natural testing algorithms for kwise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.