Sublinear-Time Computation in the Presence of Online Erasures

TL;DR

This work initiates the study of sublinear-time computation when input access is mediated by online adversarial erasures, where an oracle can erase up to $t$ values after each query. It develops online-erasure-resilient testers for linearity and quadraticity, achieving near-standard complexity for constant $t$, with a tight $\Theta(\log t)$ bound for linearity; it also proves impossibility results for sortedness and Lipschitz properties under this adversarial model. A two-player game framework and Fourier-analytic methods underpin the linearity tester, while a witness-structure approach drives the quadraticity tester, including a series of lemmas ensuring robustness to erasures. The paper further extends these ideas to online corruption and provides a generalized Yao minimax principle for online erasure/oracle settings, offering a foundation for robust sublinear algorithms under adversarial data access. Open questions include reducing the $t$-dependence in quadraticity, comparing online vs offline erasure models for new properties, and expanding corruption-resilience results to broader function classes.

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 $Θ(\log t).$ 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.

Figures (1)

• Figure 1: Stages in the quadraticity game for $t=1$, played according to the winning strategy for Player 1: connecting the $y$-decoys from the first tree to $x$-decoys (frames 1-4); drawing $z$ and connecting it to $y$-decoys and an $x$-decoy (frames 5-6), and creation of a blue triangle (frames 7--8). Frame 5 contains edges from $z$ to two structures, each replicating frame 4. We depict only points and edges relevant for subsequent frames.

Theorems & Definitions (62)

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