Abstract
The last two decades have seen enormous progress in the development of sublinear-time algorithms -- i.e., algorithms that examine/reveal properties of "data" in less time than it would take to read all of the data. A large, and important, subclass of such properties turn out to be "linear". In particular, these developments have contributed to the rich theory of probabilistically checkable proofs (PCPs) and locally testable codes (LTCs). In this survey, we focus on some of the general technical themes at work behind the many results in this area.
- Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing Reed-Muller codes. IEEE Transactions on Information Theory, 51(11):4032--4039, 2005. Google ScholarDigital Library
- Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70--122, January 1998. Google ScholarDigital Library
- László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1(1):3--40, 1991.Google ScholarCross Ref
- László Babai, Amir Shpilka, and Daniel Stefankovic. Locally testable cyclic codes. IEEE Transactions on Information Theory, 51(8):2849--2858, 2005. Google ScholarDigital Library
- Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos Kiwi, and Madhu Sudan. Linearity testing over characteristic two. IEEE Transactions on Information Theory, 42(6):1781--1795, November 1996. Google ScholarCross Ref
- Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, and Madhu Sudan. On the sum of single-orbit affine invariant properties. In preparation, 2011.Google Scholar
- Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, and Michael Viderman. Locally testable codes require redundant testers. SIAM Journal on Computing, 39(7):3230--3247, 2010. Preliminary version appeared in Proc. IEEE CCC 2009. Google ScholarDigital Library
- Eli Ben-Sasson, Prahladh Harsha, and Sofya Raskhodnikova. Some 3CNF properties are hard to test. SIAM Journal on Computing, 35(1):1--21, September 2005. Preliminary version in Proc. STOC 2003. Google ScholarDigital Library
- Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, and Madhu Sudan. Symmetric LDPC codes are not necessarily locally testable. Electronic Colloquium on Computational Complexity (ECCC), 17:199, 2010.Google Scholar
- Eli Ben-Sasson and Madhu Sudan. Robust locally testable codes and products of codes. Random Structures and Algorithms, 28(4):387--402, 2006. Google ScholarDigital Library
- Eli Ben-Sasson and Madhu Sudan. Short PCPs with polylog query complexity. SIAM J. Comput., 38(2):551--607, 2008. Preliminary version in Proc. STOC 2005. Google ScholarDigital Library
- Eli Ben-Sasson and Madhu Sudan. Limits on the rate of locally testable affine-invariant codes. Electronic Colloquium on Computational Complexity (ECCC), 17:108, 2010.Google Scholar
- Eli Ben-Sasson and Michael Viderman. Composition of semi-LTCs by two-wise tensor products. In Dinur et al. {18}, pages 378--391. Google ScholarDigital Library
- Eli Ben-Sasson and Michael Viderman. Tensor products of weakly smooth codes are robust. Theory of Computing, 5(1):239--255, 2009. Preliminary version in Proc. APPROX-RANDOM 2008. Google ScholarDigital Library
- Eli Ben-Sasson and Michael Viderman. Low rate is insufficient for local testability. Electronic Colloquium on Computational Complexity (ECCC), 17:4, 2010.Google Scholar
- Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549--595, 1993. Google ScholarDigital Library
- Irit Dinur. The PCP theorem by gap amplification. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 241--250, New York, 2006. ACM Press. Preliminary version appeared as an ECCC Technical Report TR05-046. Google ScholarDigital Library
- Irit Dinur, Klaus Jansen, Joseph Naor, and José D. P. Rolim, editors. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings, volume 5687 of Lecture Notes in Computer Science. Springer, 2009. Google ScholarDigital Library
- Irit Dinur, Madhu Sudan, and Avi Wigderson. Robust local testability of tensor products of LDPC codes. In Josep Díaz, Klaus Jansen, José D. P. Rolim, and Uri Zwick, editors, APPROX-RANDOM, volume 4110 of Lecture Notes in Computer Science, pages 304--315. Springer, 2006. Google ScholarDigital Library
- Oded Goldreich, editor. Property Testing - Current Research and Surveys {outgrow of a workshop at the Institute for Computer Science (ITCS) at Tsinghua University, January 2010}, volume 6390 of Lecture Notes in Computer Science. Springer, 2010. Google ScholarDigital Library
- Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. JACM, 45(4):653--750, 1998. Google ScholarDigital Library
- Oded Goldreich and Tali Kaufman. Proximity oblivious testing and the role of invariances. Electronic Colloquium on Computational Complexity (ECCC), 17:58, 2010.Google Scholar
- Oded Goldreich and Dana Ron. On proximity oblivious testing. In Michael Mitzenmacher, editor, STOC, pages 141--150. ACM, 2009. Google ScholarDigital Library
- Oded Goldreich and Or Sheffet. On the randomness complexity of property testing. In Moses Charikar, Klaus Jansen, Omer Reingold, and José D. P. Rolim, editors, APPROX-RANDOM, volume 4627 of Lecture Notes in Computer Science, pages 509--524. Springer, 2007. Google ScholarDigital Library
- Oded Goldreich and Madhu Sudan. Locally testable codes and PCPs of almost-linear length. Journal of the ACM, 53(4):558--655, 2006. Preliminary version in FOCS 2002. Google ScholarDigital Library
- Elena Grigorescu. Symmetries in Algebraic Property Testing. PhD thesis, MIT, August 2010. Google ScholarDigital Library
- Elena Grigorescu, Tali Kaufman, and Madhu Sudan. 2-transitivity is insufficient for local testability. In CCC 2008: Proceedings of the 23rd IEEE Conference on Computational Complexity, page (to appear). IEEE Computer Society, June 23-26th 2008. Google ScholarDigital Library
- Elena Grigorescu, Tali Kaufman, and Madhu Sudan. Succinct representation of codes with applications to testing. In Dinur et al. {18}, pages 534--547. Google ScholarDigital Library
- Charanjit S. Jutla, Anindya C. Patthak, Atri Rudra, and David Zuckerman. Testing low-degree polynomials over prime fields. In FOCS '04: Proceedings of the Forty-Fifth Annual IEEE Symposium on Foundations of Computer Science, pages 423--432. IEEE Computer Society, 2004. Google ScholarDigital Library
- Tali Kaufman and Simon Litsyn. Almost orthogonal linear codes are locally testable. In Proceedings of the Forty-sixth Annual Symposium on Foundations of Computer Science, pages 317--326, 2005. Google ScholarDigital Library
- Tali Kaufman and Shachar Lovett. Testing of exponentially large codes, by a new extension to Weil bound for character sums. Electronic Colloquium on Computational Complexity (ECCC), 17:65, 2010.Google Scholar
- Tali Kaufman and Dana Ron. Testing polynomials over general fields. SIAM J. Comput., 36(3):779--802, 2006. Google ScholarDigital Library
- Tali Kaufman and Madhu Sudan. Algebraic property testing: The role of invariance. Technical Report TR07-111, Electronic Colloquium on Computational Complexity, 2 November 2007. Extended abstract in Proc. 40th STOC, 2008. Google ScholarDigital Library
- Tali Kaufman and Madhu Sudan. Sparse random linear codes are locally decodable and testable. In FOCS, pages 590--600. IEEE Computer Society, 2007. Google ScholarDigital Library
- Tali Kaufman and Avi Wigderson. Symmetric LDPC codes and local testing. In Andrew Chi-Chih Yao, editor, ICS, pages 406--421. Tsinghua University Press, 2010.Google Scholar
- Swastik Kopparty and Shubhangi Saraf. Tolerant linearity testing and locally testable codes. In Dinur et al. {18}, pages 601--614. Google ScholarDigital Library
- Swastik Kopparty and Shubhangi Saraf. Local list-decoding and testing of random linear codes from high error. In Leonard J. Schulman, editor, STOC, pages 417--426. ACM, 2010. Google ScholarDigital Library
- Or Meir. Combinatorial construction of locally testable codes. SIAM J. Comput., 39(2):491--544, 2009. Google ScholarDigital Library
- Alexander Polishchuk and Daniel A. Spielman. Nearly linear-size holographic proofs. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 194--203, New York, NY, 23-25 May 1994. ACM Press. Google ScholarDigital Library
- Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 475--484, New York, NY, 1997. ACM Press. Google ScholarDigital Library
- Dana Ron. Algorithmic and analysis techniques in property testing. Foundations and Trends in Theoretical Computer Science, 5(2):73--205, 2009. Google ScholarDigital Library
- Ronitt Rubinfeld. Sublinear time algorithms. In Proceedings of International Congress of Mathematicians, volume III, pages 1095--1110. European Mathematical Society, 22-30 August 2006.Google Scholar
- Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252--271, April 1996. Google ScholarDigital Library
- Madhu Sudan. Invariance in property testing. In Goldreich {20}, pages 211--227.Google Scholar
- Paul Valiant. The tensor product of two codes is not necessarily robustly testable. In Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, APPROX-RANDOM, volume 3624 of Lecture Notes in Computer Science, pages 472--481. Springer, 2005. Google ScholarDigital Library
Index Terms
- Guest column: testing linear properties: some general theme
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