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Estimating the distance to a monotone function
Author(s) -
Ailon Nir,
Chazelle Bernard,
Comandur Seshadhri,
Liu Ding
Publication year - 2007
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20167
Subject(s) - monotone polygon , property testing , monotonic function , property (philosophy) , binary logarithm , mathematics , log log plot , function (biology) , interval (graph theory) , combinatorics , discrete mathematics , object (grammar) , computer science , artificial intelligence , mathematical analysis , philosophy , geometry , epistemology , evolutionary biology , biology
In standard property testing, the task is to distinguish between objects that have a property and those that are ε‐far from , for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy . This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1,…, n } ↦ R is at distance ε f from being monotone if it can (and must) be modified at ε f n places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 − δ)ε,ε] that encloses ε f . The running time of our algorithm is O (ε f −1 log log ε f − 1 log n ), which is optimal within a factor of loglog ε f −1 and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O (ε f −1 log n log log log n ). Finally, we extend our results to multivariate functions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007