Tight Bounds for Monotone Minimal Perfect Hashing

Sepehr Assadi, Martín Farach-Colton, William Kuszmaul

    Research output: Chapter in Book/Report/Conference proceedingConference contribution


    The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set S = {s1, ..., sn} of n distinct keys from a universe U of size u, create a data structure D that answers the following query: RANK(q) = (equation presented){rank of q in S arbitrary answer q ∈ Sotherwise. Solutions to the MMPHF problem are in widespread use in both theory and practice. The best upper bound known for the problem encodes D in O(n log log log u) bits and performs queries in O(log u) time. It has been an open problem to either improve the space upper bound or to show that this somewhat odd looking bound is tight. In this paper, we show the latter: any data structure (deterministic or randomized) for monotone minimal perfect hashing of any collection of n elements from a universe of size u requires Ω(n · log log log u) expected bits to answer every query correctly. We achieve our lower bound by defining a graph G where the nodes are the possible (un) inputs and where two nodes are adjacent if they cannot share the same D. The size of D is then lower bounded by the log of the chromatic number of G.

    Original languageEnglish (US)
    Title of host publication34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
    PublisherAssociation for Computing Machinery
    Number of pages21
    ISBN (Electronic)9781611977554
    StatePublished - 2023
    Event34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023 - Florence, Italy
    Duration: Jan 22 2023Jan 25 2023

    Publication series

    NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


    Conference34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

    ASJC Scopus subject areas

    • Software
    • General Mathematics


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