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Bounds for c-Ideal Hashing

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Fundamentals of Computation Theory (FCT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14292))

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Abstract

In this paper, we analyze hashing from a worst-case perspective. To this end, we study a new property of hash families that is strongly related to d-perfect hashing, namely c-ideality. On the one hand, this notion generalizes the definition of perfect hashing, which has been studied extensively; on the other hand, it provides a direct link to the notion of c-approximativity. We focus on the usually neglected case where the average load \(\alpha \) is at least 1 and prove upper and lower parametrized bounds on the minimal size of c-ideal hash families.

As an aside, we show how c-ideality helps to analyze the advice complexity of hashing. The concept of advice, introduced a decade ago, lets us measure the information content of an online problem. We prove hashing’s advice complexity to be linear in the hash table size.

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Notes

  1. 1.

    Note that the number of hash functions, \(|\mathcal {H}_\text {all}|=m^u\), is huge.

  2. 2.

    This number can be expressed as \(\sum _{k=1}^m\left( {\begin{array}{c}\alpha _k\\ 2\end{array}}\right) \in \varTheta \left( \alpha _1^2+\ldots +\alpha _m^2\right) \subseteq \mathcal {O}\left( m\cdot \alpha _\text {max}^2\right) \).

  3. 3.

    Note that, for any \(h_\textrm{eq}\), there are exactly (\(u\!\!\mod m\)) cells of size \(\lceil {u/m} \rceil \) since \(\sum _{k=1}^m |h_\textrm{eq}^{-1}(k)| = u\).

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Authors and Affiliations

  1. ETH Zürich, Zürich, Switzerland

    Fabian Frei & David Wehner

Authors
  1. Fabian Frei
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  2. David Wehner
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Correspondence to Fabian Frei .

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Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

  2. University of Kiel, Kiel, Germany

    Klaus Jansen

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Frei, F., Wehner, D. (2023). Bounds for c-Ideal Hashing. In: Fernau, H., Jansen, K. (eds) Fundamentals of Computation Theory. FCT 2023. Lecture Notes in Computer Science, vol 14292. Springer, Cham. https://doi.org/10.1007/978-3-031-43587-4_15

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  • DOI: https://doi.org/10.1007/978-3-031-43587-4_15

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  • Online ISBN: 978-3-031-43587-4

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