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Sublinear-Time Computation in the Presence of Online Erasures

Authors Iden Kalemaj , Sofya Raskhodnikova , Nithin Varma

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Author Details

Iden Kalemaj
  • Department of Computer Science, Boston University, MA, USA
Sofya Raskhodnikova
  • Department of Computer Science, Boston University, MA, USA
Nithin Varma
  • Chennai Mathematical Institute, India

Funding

  • Kalemaj, Iden: This work was supported by NSF award CCF-1909612 and Boston University’s Dean’s Fellowship.
  • Raskhodnikova, Sofya: This work was supported by NSF award CCF-1909612.
  • Varma, Nithin: This work was done in part while the author was a postdoctoral researcher at the University of Haifa, Israel, where he was supported by the Israel Science Foundation, grant number 497/17 and the PBC Fellowship for Postdoctoral Fellows by the Israeli Council of Higher Education.

Acknowledgements

We are thankful to Kobbi Nissim for suggesting investigating settings with online adversarial erasures. We thank Noga Ron-Zewi for pointing out relevant references and the anonymous ITCS reviewers for helpful comments.

Cite AsGet BibTex

Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma. Sublinear-Time Computation in the Presence of Online Erasures. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 90:1-90:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.90

Abstract

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Randomized algorithms
  • property testing
  • Fourier analysis
  • linear functions
  • quadratic functions
  • Lipschitz and monotone functions
  • sorted sequences

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