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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.33

On the Amortized Complexity of Approximate Counting

Authors: Ishaq Aden-Ali, Yanjun Han, Jelani Nelson, and Huacheng Yu

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract

Naively storing a counter up to value n would require Ω(log n) bits of memory. Nelson and Yu [Jelani Nelson and Huacheng Yu, 2022], following work of Morris [Robert H. Morris, 1978], showed that if the query answers need only be (1+ε)-approximate with probability at least 1 - δ, then O(log log n + log log(1/δ) + log(1/ε)) bits suffice, and in fact this bound is tight. Morris' original motivation for studying this problem though, as well as modern applications, require not only maintaining one counter, but rather k counters for k large. This motivates the following question: for k large, can k counters be simultaneously maintained using asymptotically less memory than k times the cost of an individual counter? That is to say, does this problem benefit from an improved amortized space complexity bound? We answer this question in the negative. Specifically, we prove a lower bound for nearly the full range of parameters showing that, in terms of memory usage, there is no asymptotic benefit possible via amortization when storing multiple counters. Our main proof utilizes a certain notion of "information cost" recently introduced by Braverman, Garg and Woodruff [Mark Braverman et al., 2020] to prove lower bounds for streaming algorithms.

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Ishaq Aden-Ali, Yanjun Han, Jelani Nelson, and Huacheng Yu. On the Amortized Complexity of Approximate Counting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{adenali_et_al:LIPIcs.APPROX/RANDOM.2024.33,
  author =	{Aden-Ali, Ishaq and Han, Yanjun and Nelson, Jelani and Yu, Huacheng},
  title =	{{On the Amortized Complexity of Approximate Counting}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.33},
  URN =		{urn:nbn:de:0030-drops-210264},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.33},
  annote =	{Keywords: streaming, approximate counting, information complexity, lower bounds}
}

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On the Amortized Complexity of Approximate Counting

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