Estimating the distance to a monotone function

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_fn 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 log log 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).