Perfect hashing and related problems

Abstract

One of the most common tasks in many computer applications is the maintenance of a dictionary of words. The three most basic operations on the dictionary are find, insert, and delete. An important data structure that supports these operations is a hash table. On a hash table, a basic operation takes 𝑂(1) time in the average case and 𝑂(𝑛) time in the worst case, where n is the number of words in the dictionary. While an ordinary hash function maps the words in a dictionary to a hash table with collisions, a perfect hash function maps the words in a dictionary to a hash table with no collisions. Thus, perfect hashing is a special case of hashing, in which a find operation takes 𝑂(1) time in the worst case, and an insert or a delete operation takes 𝑂(1) time in the average case and 𝑂(𝑛) time in the worst case.

This thesis addresses the following issues:

Mapping, ordering and searching (MOS) is a successful algorithmic approach to finding perfect hash functions for static dictionaries. Previously, no analysis has been given for the running time of the MOS algorithm. In this thesis, a lower bound is proved on the tradeoff between the time required to find a perfect hash function and the space required to represent the perfect hash function .

A new algorithm for static dictionaries called the musical chairs(MC) algorithm is developed that is based on ordering the hyperedges of a graph. It is found experimentally that the MC algorithm runs faster than the MOS algorithm in all cases for which the MC algorithm is capable of finding a perfect hash function.

A new perfect hashing algorithm is developed for dynamic dictionaries. In this algorithm, an insert or a delete operation takes 𝑂(1) time in the average case, and a find operation takes 𝑂(1) time in the worst case. The algorithm is modeled using results from queueing theory .

An ordering problem from graph theory, motivated by the hypergraph ordering problem in the Me algorithm, is proved to be NP-complete.