Practical perfect hashing in nearly optimal space
A hash function is a mapping from a key universe U to a range of integers, i.e., h: U bar right arrow {0, 1, . . . , m-1}, where m is the range's size. A perfect hash function for some set S subset of U is a hash function that is one-to-one on S. where m >= vertical bar S vertical bar. A minimal perfect hash function for some set S subset of U is a perfect hash function with a range of minimum size, i.e., m = vertical bar S vertical bar. This paper presents a construction for (minimal) perfect hash functions that combines theoretical analysis, practical performance, expected linear construction time and nearly optimal space consumption for the data structure. For n keys and m = n the space consumption ranges from 2.62n + o(n) to 3.3n + o(n) bits, and for m = 1.23n it ranges from 1.95n + o(n) to 2.7n + o(n) bits. This is within a small constant factor from the theoretical lower bounds of 1.44n bits for m = n and 0.89n bits for m = 1.23n. We combine several theoretical results into a practical solution that has turned perfect hashing into a very compact data structure to solve the membership problem when the key set S is static and known in advance. By taking into account the memory hierarchy we can construct (minimal) perfect hash functions for over a billion keys in 46 min using a commodity PC. An open source implementation of the algorithms is available at http://cmph.sf.net under the GNU Lesser General Public License (LGPL). (c) 2012 Elsevier Ltd. All rights reserved.
perfect hash functions randomized algorithms random graphs large key sets