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We study property testing with incomplete or noisy inputs. The models we consider allow for adversarial manipulation of the input, but differ in whether the manipulation can be done only offline, i.e., before the execution of the algorithm, or online, i.e., as the algorithm runs. The manipulations by an adversary can come in the form of erasures or corruptions. We compare the query complexity and the randomness complexity of property testing in the offline and online models. Kalemaj, Raskhodnikova, and Varma (Theory Comput. `23) provide properties that can be tested with a small number of queries with offline erasures, but cannot be tested at all with online erasures. We demonstrate that the two models are incomparable in terms of query complexity: we construct properties that can be tested with a constant number of queries in the online corruption model, but require querying a significant fraction of the input in the offline erasure model. We also construct properties that exhibit a strong separation between the randomness complexity of testing in the presence of offline and online adversaries: testing these properties in the online model requires exponentially more random bits than in the offline model, even when they are tested with nearly the same number of queries in both models. Our randomness separation relies on a novel reduction from randomness-efficient testers in the adversarial online model to query-efficient testers in the standard model. 

We study local filters for the Lipschitz property of real-valued functions f : V → [0,r], where the Lipschitz property is defined with respect to an arbitrary undirected graph G = (V, E ). We give nearly optimal local Lipschitz filters both with respect to ℓ1-distance and ℓ0-distance. Previous work only considered unbounded- range functions over [n]d. Jha and Raskhodnikova (SICOMP ‘13) gave an algorithm for such functions with lookup complexity exponential in d, which Awasthi et al. (ACM Trans. Comput. Theory) showed was necessary in this setting. We demonstrate that important applications of local Lipschitz filters can be accomplished with filters for functions whose range is bounded in [0,r]. For functions f : [n]d → [0,r], we achieve running time (dr log n )O (log r ) for the ℓ1-respecting filter and dO(r) polylog n for the ℓ0-respecting filter, thus circumventing the lower bound. Our local filters provide a novel Lipschitz extension that can be implemented locally. Furthermore, we show that our algorithms are nearly optimal in terms of the dependence on r for the domain {0,1}d, an important special case of the domain [n]d. In addition, our lower bound resolves an open question of Awasthi et al., removing one of the conditions necessary for their lower bound for general range. We prove our lower bound via a reduction from distribution-free Lipschitz testing and a new technique for proving hardness for adaptive algorithms. Finally, we provide two applications of our local filters to real-valued functions, with no restrictions on the range. In the first application, we use them in conjunction with the Laplace mechanism for differential privacy and noisy binary search to provide mechanisms for privately releasing outputs of black-box functions, even in the presence of malicious clients. In particular, our differentially private mechanism for arbitrary real-valued functions runs in time 2polylog min(r,nd ) and, for honest clients, has accuracy comparable to the Laplace mechanism for Lipschitz functions, up to a factor of O (log min(r,nd )). In the second application, we use our local filters to obtain the first nontrivial tolerant tester for the Lipschitz property. Our tester works for functions of the form f : {0,1}d → ℝ, makes queries, and has tolerance ratio 2.01. Our applications demonstrate that local filters for bounded-range functions can be applied to construct efficient algorithms for arbitrary real-valued functions.