Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps
We study the problem of testing unateness of functions f:{0,1}^d -> R. We give an O(d/\epsilon . log(d/\epsilon))-query nonadaptive tester and an O(d/\epsilon)-query adaptive tester and show that both testers are optimal for a fixed distance parameter \epsilon. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. We generalize our results to obtain optimal unateness testers for functions f:[n]^d -> R.
Our results establish that adaptivity helps with testing unateness of real-valued functions on domains of the form {0,1}^d and, more generally, [n]^d. This stands in contrast to the situation for monotonicity testing where there is no adaptivity gap for functions f:[n]^d -> R.
Property testing
unate and monotone functions
5:1-5:14
Regular Paper
Roksana
Baleshzar
Roksana Baleshzar
Deeparnab
Chakrabarty
Deeparnab Chakrabarty
Ramesh Krishnan S.
Pallavoor
Ramesh Krishnan S. Pallavoor
Sofya
Raskhodnikova
Sofya Raskhodnikova
C.
Seshadhri
C. Seshadhri
10.4230/LIPIcs.ICALP.2017.5
Nir Ailon and Bernard Chazelle. Information theory in property testing and monotonicity testing in higher dimension. Inf. Comput., 204(11):1704-1717, 2006.
Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri. Optimal unateness testers for real-valued functions: Adaptivity helps. Electronic Colloquium on Computational Complexity (ECCC), 2017, 2017. Also appeared as arXiv report 1703.05199 (https://arxiv.org/abs/1703.05199). URL: https://eccc.weizmann.ac.il/report/2017/049/.
https://eccc.weizmann.ac.il/report/2017/049/
Tugkan Batu, Ronitt Rubinfeld, and Patrick White. Fast approximate PCPs for multidimensional bin-packing problems. Inf. Comput., 196(1):42-56, 2005.
Aleksandrs Belovs and Eric Blais. Quantum algorithm for monotonicity testing on the hypercube. Theory of Computing, 11:403-412, 2015.
Aleksandrs Belovs and Eric Blais. A polynomial lower bound for testing monotonicity. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 1021-1032, 2016.
Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 164-173, 2014.
Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. SIAM J. Comput., 41(6):1380-1425, 2012.
Eric Blais, Joshua Brody, and Kevin Matulef. Property testing lower bounds via communication complexity. Computational Complexity, 21(2):311-358, 2012.
Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In Proceedings, IEEE Conference on Computational Complexity (CCC), pages 309-320, 2014.
Jop Briët, Sourav Chakraborty, David García-Soriano, and Arie Matsliah. Monotonicity testing and shortest-path routing on the cube. Combinatorica, 32(1):35-53, 2012.
Deeparnab Chakrabarty. Monotonicity testing. In Encyclopedia of Algorithms, pages 1352-1356. Springer, 2016.
Deeparnab Chakrabarty, Kashyap Dixit, Madhav Jha, and C. Seshadhri. Property testing on product distributions: Optimal testers for bounded derivative properties. In Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1809-1828, 2015.
Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and Lipschitz testing over hypercubes and hypergrids. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 419-428, 2013.
Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014.
Deeparnab Chakrabarty and C. Seshadhri. An o(n) monotonicity tester for boolean functions over the hypercube. SIAM J. Comput., 45(2):461-472, 2016.
Xi Chen, Anindya De, Rocco A. Servedio, and Li-Yang Tan. Boolean function monotonicity testing requires (almost) O(n^1/2) non-adaptive queries. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 519-528, 2015.
Xi Chen, Rocco A. Servedio, and Li-Yang Tan. New algorithms and lower bounds for monotonicity testing. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pages 286-295, 2014.
Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin M. Varma. Erasure-resilient property testing. In Proceedings, International Colloquium on Automata, Languages and Processing (ICALP), pages 91:1-91:15, 2016.
Yevgeny Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. Proceedings, International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), pages 97-108, 1999.
Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. J. Comput. System Sci., 60(3):717-751, 2000.
Eldar Fischer. On the strength of comparisons in property testing. Inf. Comput., 189(1):107-116, 2004.
Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 474-483, 2002.
Oded Goldreich. Introduction to property testing (working draft), 2015. URL: http://www.wisdom.weizmann.ac.il/~oded/PDF/pt-v1.pdf.
http://www.wisdom.weizmann.ac.il/~oded/PDF/pt-v1.pdf
Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20:301-337, 2000.
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998.
Shirley Halevy and Eyal Kushilevitz. Distribution-free property-testing. SIAM J. Comput., 37(4):1107-1138, 2007.
Shirley Halevy and Eyal Kushilevitz. Testing monotonicity over graph products. Random Struct. Algorithms, 33(1):44-67, 2008.
Madhav Jha and Sofya Raskhodnikova. Testing and reconstruction of Lipschitz functions with applications to data privacy. SIAM J. Comput., 42(2):700-731, 2013.
Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and boolean isoperimetric type theorems. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pages 52-58, 2015.
Subhash Khot and Igor Shinkar. An Õ(n) queries adaptive tester for unateness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, pages 37:1-37:7, 2016.
Eric Lehman and Dana Ron. On disjoint chains of subsets. J. Combin. Theory Ser. A, 94(2):399-404, 2001.
Sofya Raskhodnikova. Testing if an array is sorted. In Encyclopedia of Algorithms, pages 2219-2222. Springer, 2016.
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996.
Andrew Chi-Chih Yao. Probabilistic computations: Toward a unified measure of complexity (extended abstract). In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pages 222-227, 1977.
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