文章基本信息

标题：Adaptive Lower Bound for Testing Monotonicity on the Line
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作者：Aleksandrs Belovs
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2018
卷号：116
页码：1-10
DOI：10.4230/LIPIcs.APPROX-RANDOM.2018.31
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of epsilon-testing monotonicity of a function f : [n]->[r]. All our lower bounds are for adaptive two-sided testers. - We prove a nearly tight lower bound for this problem in terms of r. The bound is Omega((log r)/(log log r)) when epsilon = 1/2. No previous satisfactory lower bound in terms of r was known. - We completely characterise query complexity of this problem in terms of n for smaller values of epsilon. The complexity is Theta(epsilon^{-1} log (epsilon n)). Apart from giving the lower bound, this improves on the best known upper bound. Finally, we give an alternative proof of the Omega(epsilon^{-1}d log n - epsilon^{-1}log epsilon^{-1}) lower bound for testing monotonicity on the hypergrid [n]^d due to Chakrabarty and Seshadhri (RANDOM'13).
关键词：property testing; monotonicity on the line; monotonicity on the hypergrid