Abstract
We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σn ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is ε-far from being monotone (i.e., every monotone function differs from f on more than an ε fraction of the domain). For any ε > 0, the query complexity of the test is O((n/ε) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was \(\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)\).
We also present an alternative test for the boolean range Ξ = {0,1} whose query complexity O(n 2/ε 2 ) is independent of alphabet size ∣Σ∣.
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Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A. (1999). Improved Testing Algorithms for Monotonicity. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_10
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DOI: https://doi.org/10.1007/978-3-540-48413-4_10
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