TY - GEN

T1 - Tight Bounds for Monotone Minimal Perfect Hashing

AU - Assadi, Sepehr

AU - Farach-Colton, Martín

AU - Kuszmaul, William

N1 - Publisher Copyright:
Copyright © 2023 by SIAM.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

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M3 - Conference contribution

AN - SCOPUS:85153593572

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 456

EP - 476

BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

PB - Association for Computing Machinery

T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

Y2 - 22 January 2023 through 25 January 2023

ER -