Asymptotic distribution for the cost of linear probing hashing

Abstract

Abstract. We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola. The average cost of unsuccessful searches is considered too.

Individual displacements for linear probing hashing with different insertion policies

“…As has been noted earlier, see e.g. [10], the asymptotic behaviour of hashing with linear probing when n and m and tend to ∞ depend on the relative size of m and n. We will in the present paper, for simplicity as well as for lack of time and space, only consider the case n/m → α with 0 < α < 1. (This is also the range of most interest for computer applications.)…”

“…(It can be treated by similar methods, cf. the discussion of the total displacement in [10].) It will be seen below, that in this range, the dependency on T is negligible.…”

“…We began our study of hashing with linear probing in [10], where we studied the total displacement i d i . In the present paper, we will study the individual displacements.…”

Poisson-Kingman partitions

“…This formula for m = 2 was derived in another way by Janson [25,Theorem 7.4]. There the distribution of S 2 (λ) appears as the asymptotic distribution, in a suitable limit regime, of the cost of linear probing hashing.…”

“…Instances and variants of this model have found applications in the diverse fields of population genetics [17,19], combinatorics [4,48], Bayesian statistics [23], ecology [37,15], statistical physics [11,12,13,53,55], and computer science [25].…”