Tight Bounds for Monotone Minimal Perfect Hashing

Abstract

The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set \(S=\{s_{1},\ldots,s_{n}\}\) of \(n\) distinct keys from a universe \(U\) of size \(u\) , create a data structure \(\text{D}\) that answers the following query: \(\begin{equation*} \rm{R\small{ANK}}(q) = \begin{cases} \text{rank of } q \text{ in } S & q\in S\\ \text{arbitrary answer} & \text{otherwise.}\end{cases}\end{equation*}\) Solutions to the MMPHF problem are in widespread use in both theory and practice. The best upper bound known for the problem encodes \(\text{D}\) 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 \(\Omega(n\cdot\log\log\log{u})\) expected bits to answer every query correctly. We achieve our lower bound by defining a graph \(\mathbf{G}\) where the nodes are the possible \({u\choose n}\) inputs and where two nodes are adjacent if they cannot share the same \(\text{D}\) . The size of \(\text{D}\) is then lower bounded by the log of the chromatic number of \(\mathbf{G}\) . Finally, we show that the fractional chromatic number (and hence the chromatic number) of \(\mathbf{G}\) is lower bounded by \(2^{\Omega(n\log\log\log u)}\) .