Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Journal of Combinatorial Theory Series A
Monotonicity testing over general poset domains
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Improved Testing Algorithms for Monotonicity
RANDOM-APPROX '99 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems: Randomization, Approximation, and Combinatorial Algorithms and Techniques
On the strength of comparisons in property testing
Information and Computation
Tolerant property testing and distance approximation
Journal of Computer and System Sciences
Information theory in property testing and monotonicity testing in higher dimension
Information and Computation
Estimating the distance to a monotone function
Random Structures & Algorithms
Testing monotonicity over graph products
Random Structures & Algorithms
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Fast approximate PCPs for multidimensional bin-packing problems
Information and Computation
Lower bounds for local monotonicity reconstruction from transitive-closure spanners
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Proceedings of the 19th ACM SIGSOFT symposium and the 13th European conference on Foundations of software engineering
Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Calibrating noise to sensitivity in private data analysis
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Property Testing Lower Bounds via Communication Complexity
Computational Complexity - Selected papers from the 26th Annual IEEE Conference on Computational Complexity (CCC 2011)
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The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f:[k]n - R (for some ordered range R). The hypergrid/cube has a natural partial order given by coordinate-wise ordering, denoted by prec. A function is monotone if for all pairs x prec y, f(x) ≤ f(y). The distance to monotonicity, εf, is the minimum fraction of values of f that need to be changed to make f monotone. For k=2 (the boolean hypercube), the usual tester is the edge tester, which checks monotonicity on adjacent pairs of domain points. It is known that the edge tester using O(ε-1n log|R|) samples can distinguish a monotone function from one where εf ε. On the other hand, the best lower bound for monotonicity testing over general R is Ω(n). We resolve this long standing open problem and prove that O(n/ε) samples suffice for the edge tester. For hypergrids, known testers require O(ε-1n log k log |R|) samples, while the best known (non-adaptive) lower bound is Ω(ε-1 n log k). We give a (non-adaptive) monotonicity tester for hypergrids running in O(ε{-1} n log k) time. Our techniques lead to optimal property testers (with the same running time) for the natural Lipschitz property on hypercubes and hypergrids. (A c-Lipschitz function is one where |f(x) - f(y)| ≤ c||x-y||1.) In fact, we give a general unified proof for O(ε-1nlog k)-query testers for a class of "bounded-derivative" properties, a class containing both monotonicity and Lipschitz.