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Finding Missing Items Requires Strong Forms of Randomness

Authors Amit Chakrabarti , Manuel Stoeckl

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LIPIcs.CCC.2024.28.pdf
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Author Details

Amit Chakrabarti
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Manuel Stoeckl
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA

Funding

Supported in part by the National Science Foundation under Award 2006589.

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Amit Chakrabarti and Manuel Stoeckl. Finding Missing Items Requires Strong Forms of Randomness. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.28

Abstract

Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen as a function of earlier algorithm outputs. As with classic streaming algorithms, which must only be correct for the worst-case fixed stream, adversarially robust algorithms with access to randomness can use significantly less space than deterministic algorithms. We prove that for the Missing Item Finding problem in streaming, the space complexity also significantly depends on how adversarially robust algorithms are permitted to use randomness. (In contrast, the space complexity of classic streaming algorithms does not depend as strongly on the way randomness is used.) For Missing Item Finding on streams of length 𝓁 with elements in {1,…,n}, and ≀ 1/poly(𝓁) error, we show that when 𝓁 = O(2^√{log n}), "random seed" adversarially robust algorithms, which only use randomness at initialization, require 𝓁^Ξ©(1) bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use O(polylog(𝓁)) random bits. When 𝓁 is between n^Ξ©(1) and O(√n), "random tape" adversarially robust algorithms need 𝓁^Ξ©(1) space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use O(polylog(𝓁)) space. The space lower bound for the "random seed" case follows, by a reduction given in prior work, from a lower bound for pseudo-deterministic streaming algorithms given in this paper.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Sketching and sampling
  • Theory of computation β†’ Lower bounds and information complexity
  • Theory of computation β†’ Pseudorandomness and derandomization
Keywords
  • Data streaming
  • lower bounds
  • space complexity
  • adversarial robustness
  • derandomization
  • sketching
  • sampling

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