Estimating the Distance to a Monotone Function

Abstract

In standard property testing, the task is to distinguish between objects that have a property $\mathcal{P}$ and those that are ε-far from $\mathcal{P}$, 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 $\mathcal{P}$. 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}loglog 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 logloglog n).

This work was supported in part by NSF grants CCR-998817, CCR-0306283, ARO Grant DAAH04-96-1-0181.