## Property Testing (2000)

Venue: | Handbook of Randomized Computing, Vol. II |

Citations: | 78 - 11 self |

### BibTeX

@INPROCEEDINGS{Ron00propertytesting,

author = {Dana Ron},

title = {Property Testing},

booktitle = {Handbook of Randomized Computing, Vol. II},

year = {2000},

pages = {597--649},

publisher = {Kluwer Academic Publishers}

}

### Years of Citing Articles

### OpenURL

### Abstract

this technical aspect (as in the bounded-degree model the closest graph having the property must have at most dN edges and degree bound d as well).

### Citations

1694 | A Theory of the Learnable
- Valiant
- 1984
(Show Context)
Citation Context ...is, the distance between functions g and h is Pr xD [g(x) 6= h(x)]. The above definition of property testing is inspired by the Probably Approximately Correct (PAC) learning model proposed by Valiant =-=[Val84]-=-. In the PAC model, a learning algorithm is given a random sample labeled by an unknown function f as defined above, and is required to output (with high probability) a hypothesis h that approximates ... |

1687 | The probabilistic method - Alon, Erdős, et al. - 1992 |

647 | Some optimal inapproximability results - Hastad - 1997 |

424 | Property testing and its connection to learning and approximation
- Goldreich, Goldwasser, et al.
- 1998
(Show Context)
Citation Context ...ike a fast, low-data test that informs us whether this investment would be worthwhile. 1.2 Testing Graph Properties The study of testing graph properties was initiated by Goldreich Goldwasser and Ron =-=[GGR98]-=-. As noted previously, the precise definition of testing graph properties is dependent on the repre4 In the agnostic model [KSS94], nothing is assumed about f , and so the distance between h and f is ... |

403 | Non-deterministic exponential time has two-prover interactive protocols - Babai, Fortnow, et al. - 1991 |

345 | Self-testing/correcting with applications to numerical problems
- Blum, Luby, et al.
- 1993
(Show Context)
Citation Context ... of functions (properties), testing can be done much more efficiently than learning (in terms of sample/query complexity and/or running time). This is true for example in the case of linear functions =-=[BLR93]-=-, multivariate polynomials [RS96], and monotone functions [GGL + 00]. If we have fast testing algorithms that require relatively small samples, we may be able to use such algorithms in the context of ... |

328 | Robust characterization of polynomials with applications to program testing
- Rubinfeld, Sudan
- 1996
(Show Context)
Citation Context ...rithm for the latter problem can be used to solve the former problem. 1.1.3 Property Testing, Program Testing and PCP Property testing of functions was first explicitly defined by Rubinfeld and Sudan =-=[RS96]-=- in the context of program testing . The goal of a program testing algorithm is to test whether a given program computes a specified function. Here one may choose to test that the program satisfies a ... |

282 | Safra,A sub-constant error-probability low-degree test and sub-constant error-probability PCP characterization of NP - Raz, S - 1997 |

258 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...in the context of probabilistically checkable proofs (pcp). In this context the property being tested is whether the function is a codeword of a specific code. This paradigm, explicitly introduced in =-=[BFLS91]-=-, has shifted from testing codes defined by 3 low-degree polynomials [BFL91, BFLS91, FGL + 96, AS98, ALM + 98] to testing Hadamard codes [ALM + 98, BGLR93, BS94, BCH + 95, Kiw96, Tre98], and to testin... |

206 | Free bits, PCPs and non-approximability — towards tight results - Bellare, Goldreich, et al. - 1998 |

194 | Toward efficient agnostic learning
- Kearns, Schapire, et al.
- 1994
(Show Context)
Citation Context ...esting graph properties was initiated by Goldreich Goldwasser and Ron [GGR98]. As noted previously, the precise definition of testing graph properties is dependent on the repre4 In the agnostic model =-=[KSS94]-=-, nothing is assumed about f , and so the distance between h and f is required to be not much larger than the distance between f and the closest function in H. 4 sentation of graphs. There are two sta... |

176 | Using dual approximation algorithms for scheduling problems: Theoretical and practical results - Hochbaum, Shmoys - 1987 |

171 | Efficient probabilistically checkable proofs and applications to approximations - Bellare, Goldwasser, et al. - 1993 |

158 | Efficient testing of large graphs
- Alon, Fischer, et al.
- 2000
(Show Context)
Citation Context ...class of properties, presented in [GGR98], has query complexity i ~ O(k 2 )=ffl j 2k+8 and running time exp i ~ O(k 2 )=ffl j k+1 . First Order Graph Properties Alon, Fischer, Krivelevich and Szegedy =-=[AFKS99]-=-, study the class of first order graph properties. These are properties that can be formulated by first order expressions about graphs. That is, expressions that contain quantifiers over vertices, boo... |

146 | Improved low-degree testing and its applications - Arora, Sudan |

126 |
Regular partitions of graphs
- Szemerédi
- 1978
(Show Context)
Citation Context ...g algorithm for k-Colorability whose complexity is independent of N was already implicit in work of Alon et. al. [ADL + 94]. They build on a constructive version of the Regularity Lemma of Szemer'edi =-=[Sze78]-=- which they prove, and the complexity of the resulting testing algorithm is a tower of poly(1=ffl) exponents. Constructing (Good) Partitions For all the above properties, in case the graph has the des... |

120 | Property testing in bounded degree graphs
- Goldreich, Ron
(Show Context)
Citation Context ...acency-matrix model (Definition 2.1.2), with the appropriate notions of distance. 8 Once again, for sake of simplicity, this definition slightly differs from that discussed in the introduction and in =-=[GR97]-=-. There, distance is defined as the fraction of entries in the N \Theta d matrix representation on which the two graphs differ. According to that definition, each (undirected) edge (v; u) in the symme... |

117 | Improved non-approximability results - Bellare, Sudan - 1994 |

113 | Quick approximation to matrices and applications
- Frieze, Kannan
- 1999
(Show Context)
Citation Context ...ffl)N 2 crossing edges. The basic idea is that the partition of the sample that caused the algorithm to accept is used to partition the whole graph. This idea was later also used by Frieze and Kannan =-=[FK99]-=- to obtain polynomial time approximation schemes for various problems, where they apply a relaxed constructive version of Szemer'edi's Regularity Lemma [Sze78]. General Graph Partition Properties All ... |

91 | The algorithmic aspects of the regularity lemma - Alon, Duke, et al. - 1994 |

83 | sublinear bipartite tester for bounded degree graphs
- Goldreich, A
- 1999
(Show Context)
Citation Context ...s on H can be related to events occurring in the real walks on G. This is done by defining a special Markov chain given H and relating walks on the Markov chain to walks on G. For further details see =-=[GR99]-=-. 3.5 Directed Graphs As noted at the end of Subsection 2.2, for some properties of undirected graphs that have analogous properties in directed graphs, algorithms that work on undirected graphs can b... |

78 | Regular languages are testable with a constant number of queries
- Alon, Krivelevich, et al.
- 1999
(Show Context)
Citation Context ...ative. 11 . This assumption can be removed at a further multiplicative cost of O( p G). 4.2 Testing Regular Languages In this subsection we sketch the result of Alon, Krivelevich, Newman, and Szegedy =-=[AKNS99]-=-, showing that for every regular language L ` f0; 1g , there exists a testing algorithm for L. Namely, the algorithm accepts every word w 2 L, and rejects with probability at least 2=3 every word w th... |

61 |
Conductance and convergence of Markov chains: a combinatorial treatment of expanders
- Mihail
(Show Context)
Citation Context ... probability that differs significantly from 1=N . In other words, the distribution on the ending vertices (when starting from s) is far from stationary. What can be shown (using techniques of Mihail =-=[Mih89]-=-) is that this implies the existence of a small cut between some set of vertices S that are each reached 27 from s with probability that is roughly 1= p jSj \Delta N and the rest of G. Furthermore, it... |

57 | Probabilistic checkable proofs: A new characterization of np - Arora, Safra - 1998 |

53 |
On recognizing graph properties from adjacency matrices
- Rivest, Vuillemin
- 1976
(Show Context)
Citation Context ...ose query complexities do not depend at all on the size of the graph. This should be put in contrast to known lower bounds on the complexity of exactly deciding graph properties. Rivest and Vuillemin =-=[RV76]-=- showed that any deterministic procedure for deciding any non-trivial monotone N-vertex graph property must examine\Omega\Gamma N 2 ) entries in the adjacency matrix representing the graph, thus resol... |

50 | Testing k-colorability
- Alon, Krivelevich
(Show Context)
Citation Context ...t exp(poly(1=ffl)). In what follows, N denotes the number of graph vertices. ffl Bipartiteness. The algorithm has query complexity and running time ~ O(ffl \Gamma3 ). 6 Recently, Alon and Krivelevich =-=[AK99]-=- improved the analysis of the algorithm and obtained a bound of ~ O(ffl \Gamma2 ) on the query complexity and running time. ffl k-colorability, ks3. The algorithm has query complexity ~ O \Gamma k 4 =... |

50 | Global Min-Cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm
- Karger
- 1993
(Show Context)
Citation Context ... above. The procedure for detecting whether a vertex v belongs to a k-connected subset C of size at most n such that the cut (C; V n C) has size k \Gamma 1, is based on Karger's Contraction Algorithm =-=[Kar93]-=- which is a randomized algorithm for finding a minimum cut in a graph. Given a vertex v and a size bound n, the following randomized search process is performed \Theta(n 2\Gamma 2 k ) times, or until ... |

35 | Testing of the long code and hardness for clique - H˚astad - 1996 |

27 | Testing the diameter of graphs
- Parnas, Ron
- 2002
(Show Context)
Citation Context ...e again, the factor of 1 2 is because each edge (u; v) is represented both as an entry [u; i] and as an entry [v; j]). A variant of the above model allows the incidence lists to be of varying lengths =-=[PR99a]-=-. In such a case, the distance between graphs is defined with respect to the total number of edges in the graph (or an upper bound on this number). This model is suitable for testing graphs that are n... |

26 |
On the time required to recognize properties of graphs: a problem
- Rosenberg
- 1973
(Show Context)
Citation Context ... for deciding any non-trivial monotone N-vertex graph property must examine\Omega\Gamma N 2 ) entries in the adjacency matrix representing the graph, thus resolving the Aanderaa--Rosenberg Conjecture =-=[Ros73]-=-. The query complexity of randomized decision procedures was conjectured by Yao to be also\Omega\Gamma N 2 ). Progress towards proving this conjecture was made by Yao [Yao87], King [Kin91] and Hajnal ... |

25 | A polynomial approximation scheme for machine scheduling on uniform processors: Using the dual approximation approach - Hochbaum, Shmoys - 1988 |

25 | Recycling Queries in PCPs and in Linearity Tests - Trevisan - 1998 |

23 | An Ω(n 4/3 ) lower bound on the randomized complexity of graph properties
- Hajnal
- 1991
(Show Context)
Citation Context .... The query complexity of randomized decision procedures was conjectured by Yao to be also\Omega\Gamma N 2 ). Progress towards proving this conjecture was made by Yao [Yao87], King [Kin91] and Hajnal =-=[Haj91]-=- culminating in an \Omega\Gamma N 4=3 ) lower bound. 1.1.1 A Tradeoff Between Accuracy and Efficiency It follows from the above discussion that Property Testing trades accuracy for efficiency , where ... |

22 | Queries. Testing of function that have small width branching programs
- Newman, A
- 2000
(Show Context)
Citation Context ...is, independent of the length n of w. (The running time is dependent on the size of the (smallest) finite automaton accepting M, but this size is a fixed constant with respect to n). Recently, Newman =-=[New00]-=- extended this result and gave an algorithm having query complexity poly(1=ffl) for testing whether a word w is accepted by a given constant-width branching program. We note that Alon et. al. also sho... |

21 |
Robust functional equations and their applications to program testing
- Rubinfeld
- 1999
(Show Context)
Citation Context ...es on x d+1 ](8)s1 \Gamma ffi (9) and so the probability that such a polynomial q does not exist is at least ffi as claimed. 30 4.1.3 Testing Other Algebraic Properties Functional Equations Rubinfeld =-=[Rub99]-=- studies properties of functions f : X ! Y that can be characterized by (quantified) functional equations of the form: 8x; y 2 X, F [f(x); f(y); f(x + y); f(x \Gamma y)] = 0. For example, linearity fa... |

18 |
Lower bounds to randomized algorithms for graph properties
- Yao
(Show Context)
Citation Context ...raa--Rosenberg Conjecture [Ros73]. The query complexity of randomized decision procedures was conjectured by Yao to be also\Omega\Gamma N 2 ). Progress towards proving this conjecture was made by Yao =-=[Yao87]-=-, King [Kin91] and Hajnal [Haj91] culminating in an \Omega\Gamma N 4=3 ) lower bound. 1.1.1 A Tradeoff Between Accuracy and Efficiency It follows from the above discussion that Property Testing trades... |

16 | Testing problems with sub-learning sample complexity
- Kearns, Ron
(Show Context)
Citation Context ...ing with queries is more efficient, there are still efficient testing algorithms that use only random examples. This is in particular true of testing for decision trees over [0; 1] d (for constant d) =-=[KR98]-=-. That is, the property is belonging to the class of decisions trees over [0; 1] d having at most s nodes. This class of functions is defined as follows. Given an input ~x = (x 1 ; : : : ; x d ), the ... |

15 | Probabilistically Checkable Proofs and the Testing of Hadamard-like Codes - Kiwi - 1996 |

13 | Testing acyclicity of directed graphs in sublinear time
- Bender, Ron
- 2000
(Show Context)
Citation Context ...ber of edges in the graph by sampling. This is not true of directed graphs. Namely, a directed graph may be very dense but still acyclic. Hence, for this property a non-trivial analysis is needed. In =-=[BR00] it is sho-=-wn that the "natural" algorithm that takes a sample of vertices and accepts or rejects based on the acyclicity of the induced subgraphs, is a testing algorithm. The required sample size is O... |

11 | Maintaining the classes of 4-edge-connectivity in a graph on-line
- Dinitz, Westbrook
- 1998
(Show Context)
Citation Context ...of size less than k. Similarly to the k = 1 case, it can be shown that if a graph is ffl-far from being k--connected then it has many such components. This can be shown by defining an auxiliary graph =-=[DW98]-=- whose nodes are components of the graph and that is based on the cactus structure of [DKL76]. In addition, there are efficient procedures for recognizing such a component given a vertex that resides ... |

7 |
On the structure of the system of minimum edge cuts in a graph
- Dinic, Karazanov, et al.
- 1976
(Show Context)
Citation Context ... from being k--connected then it has many such components. This can be shown by defining an auxiliary graph [DW98] whose nodes are components of the graph and that is based on the cactus structure of =-=[DKL76]-=-. In addition, there are efficient procedures for recognizing such a component given a vertex that resides in it. In what follows we sketch these procedures, for the different values of k. For simplic... |

7 |
Private communications
- Sudan
- 1995
(Show Context)
Citation Context ...s ffl-close to P. 4.1 Testing Algebraic Properties 4.1.1 Testing Linearity In this subsection we present a test due to Blum, Luby and Rubinfeld [BLR93], with a slightly modified analysis due to Sudan =-=[Sud99]-=-. Definition 4.1.1 (linearity of a function) Let F be a finite field. A function f : F m ! F is called linear (or more precisely, multi-linear) if there exist constants a 1 ; : : : ; am 2 F s.t. for a... |

4 |
An \Omega\Gamma n 5=4 ) lower bound on the randomized complexity of graph properties
- King
- 1991
(Show Context)
Citation Context ... Conjecture [Ros73]. The query complexity of randomized decision procedures was conjectured by Yao to be also\Omega\Gamma N 2 ). Progress towards proving this conjecture was made by Yao [Yao87], King =-=[Kin91]-=- and Hajnal [Haj91] culminating in an \Omega\Gamma N 4=3 ) lower bound. 1.1.1 A Tradeoff Between Accuracy and Efficiency It follows from the above discussion that Property Testing trades accuracy for ... |

2 |
Testing the diameter of graphs. Available from http://www.eng.tau.ac.il/~danar
- Parnas, Ron
- 1999
(Show Context)
Citation Context ... vertex to any other vertex. The algorithm of [GR97] presented in Subsection 3.3 can be extended to the directed case if the algorithm can also perform queries about the incoming edges to each vertex =-=[PR99b]-=-. Otherwise, (the algorithm can only perform queries about outgoing edges), a simple lower bound of p N on the number of queries can be obtained. In the case of testing acyclicity of directed graphs t... |