🤖 AI Summary
This paper studies sublinear-time property testing under the online adversarial erasure model, where an adversary may erase up to $t$ input positions after each query, dynamically obstructing data access. We formally define this online erasure query model and establish a systematic testing framework integrating probabilistic verification, adaptive querying, and structured violation analysis. Our main contributions are: (1) proving that linearity and quadraticity are efficiently testable with tight query complexity $Theta(log t)$, matching the non-erasure setting for constant $t$; (2) demonstrating that fundamental properties such as sortedness are inherently untestable under online erasures; and (3) providing the first characterization of the intrinsic vulnerability of multiple property classes to dynamic data corruption, thereby laying a theoretical foundation for designing robust sublinear algorithms resilient to online adversarial interference.
📝 Abstract
We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each input query, such an oracle can erase $t$ input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm's access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant $t$ with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of $t$, showing that the query complexity is $Theta(log t).$ In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for $t=1$. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties. We also consider implications of our results for algorithms that are resilient to online adversarial corruptions instead of erasures.
