A Note on the Efficiency of Hashing Functions

Abstract

Hashing functions are considered which store data “randomly” in a fixed size hash table with no auxiliary storage. It is known that if k out of the N locations have been previously filled randomly, then the expected number of locations which must be looked at until an empty one is found is [formula omitted]. It is shown that there exist nonrandom hashing functions which are more efficient for certain values of k and N. However, it is shown that the 1 + k/(N - k + 1) figure is a “lower bound” on hashing performance in the following sense. If h is any hashing function such that the expected number of trials to insert the (k -b 1)st item into a table of size N is C(k, N), and if for some [formula omitted], then there exists [formula omitted]. Finally, it is shown that randomness is a sufficient but not necessary condition for attainment of the [formula omitted] performance for all k, and a necessary and sufficient condition is given. © 1972, ACM. All rights reserved.