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Volume 19 (2023) Article 1 pp. 1-48
Sublinear-Time Computation in the Presence of Online Erasures
Received: June 13, 2022
Revised: June 29, 2023
Published: August 23, 2023
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Keywords: randomized algorithms, property testing, Fourier analysis, linear functions, quadratic functions, Lipschitz and monotone functions, sorted sequences
ACM Classification: F.2
AMS Classification: 68Q25

Abstract: [Plain Text Version]

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.


A preliminary version of this paper appeared in the Proceedings of the 13th Innovations in Theoretical Computer Science Conference (ITCS'22).