Tight Bounds for Monotone Minimal Perfect Hashing

The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set S= s1,ldots,sn of n distinct keys from a universe U of size u, create a data structure DS that answers the following query: [ RankOp(q) = rank of q in S for all q∈ S ~ and arbitrary answer otherwise. ] Solutions to the MMPHF problem are in widespread use in both theory and practice. The best upper bound known for the problem encodes DS in O(nlogloglog 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: specifically that 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 · logloglogu) expected bits to answer every query correctly. We achieve our lower bound by defining a graph G where the nodes are the possible u choose n inputs and where two nodes are adjacent if they cannot share the same DS. The size of DS is then lower bounded by the log of the chromatic number of G. Finally, we show that the fractional chromatic number (and hence the chromatic number) of G is lower bounded by 2Omega(n logloglog u).