Erasure-Resilient Property Testing

Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function via an oracle that returns function values at all queried domain points. In many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. We initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester for a property P is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of the function values have been erased. The tester does not know whether a point is erased unless it queries that point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies P. It has to reject with high probability if, for all assignments of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the nonerased domain points to satisfy P. We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by using standard testers as a black box. But there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several classes of such properties including monotonicity, the Lipschitz property, and convexity. Finally, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size.