Simplified Tight Bounds for Monotone Minimal Perfect Hashing

Author Dmitry Kosolobov

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Dmitry Kosolobov
  • Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg, Russia

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Dmitry Kosolobov. Simplified Tight Bounds for Monotone Minimal Perfect Hashing. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Given an increasing sequence of integers x₁,…,x_n from a universe {0,…,u-1}, the monotone minimal perfect hash function (MMPHF) for this sequence is a data structure that answers the following rank queries: rank(x) = i if x = x_i, for i ∈ {1,…,n}, and rank(x) is arbitrary otherwise. Assadi, Farach-Colton, and Kuszmaul recently presented at SODA'23 a proof of the lower bound Ω(n min{log log log u, log n}) for the bits of space required by MMPHF, provided u ≥ n 2^{2^{√{log log n}}}, which is tight since there is a data structure for MMPHF that attains this space bound (and answers the queries in O(log u) time). In this paper, we close the remaining gap by proving that, for u ≥ (1+ε)n, where ε > 0 is any constant, the tight lower bound is Ω(n min{log log log u/n, log n}), which is also attainable; we observe that, for all reasonable cases when n < u < (1+ε)n, known facts imply tight bounds, which virtually settles the problem. Along the way we substantially simplify the proof of Assadi et al. replacing a part of their heavy combinatorial machinery by trivial observations. However, an important part of the proof still remains complicated. This part of our paper repeats arguments of Assadi et al. and is not novel. Nevertheless, we include it, for completeness, offering a somewhat different perspective on these arguments.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • monotone minimal perfect hashing
  • lower bound
  • hash


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