Sublinear-Time Computation in the Presence of Online Erasures

TL;DR

This paper explores the limits of sublinear-time algorithms under online adversarial erasures, demonstrating that certain properties like linearity and quadraticity remain testable with similar complexity, while others cannot be tested at all.

Contribution

It introduces a new model for property testing with online erasures and establishes tight bounds for testing linearity and quadraticity under this adversarial setting.

Findings

Linearity testing remains feasible with query complexity Θ(log t).

Quadraticity testing is also achievable with similar complexity.

Some properties, such as sortedness and Lipschitz continuity, are untestable even with minimal erasures.

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.