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. 1.

. Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 X i of some other variables, where X i and Y i are dependent but the pairs (X i ; Y i ) form an i.i.d. sequence. We prove, for a triangular array (X ni ; Y ni ) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes. 1. Introduction Many random variables arising in different areas of probability theory, combinatorics and statistics turn out to have the same distribution as a sum of independent random variables conditioned on a specific value of another such sum. More precisely, we are concerned with variables with the distribution of P N 1 Y i conditioned on P N 1 X...

Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.

1. Simply generated trees and Galton–Watson trees We suppose that we are given a fixed weight sequence w = (wk)k�0 of non-negative real numbers. We then define the weight of a finite rooted and ordered (a.k.a. plane) tree T by w(T): = ∏ wd +(v), (1.1) v∈T taking the product over all nodes v in T, where d + (v) is the outdegree of v. Trees with such weights are called simply generated trees and were introduced by Meir and Moon [24]. We let Tn be the random simply generated tree obtained by picking a tree with n nodes at random with probability proportional to its weight. (To avoid trivialities, we assume that w0> 0 and that there exists some k � 2 with wk> 0. We consider only n such that there exists some tree with n vertices and positive weight.) One particularly important case is when ∑ ∞ k=0 wk = 1, so the weight sequence (wk) is a probability distribution on Z�0. (We then say that (wk) is a probability weight sequence.) In this case we let ξ be a random variable with the corresponding distribution: P(ξ = k) = wk. It is easily seen that the simply generated random tree Tn equals the conditioned Galton–Watson tree with offspring distribution ξ, i.e., the random Galton–Watson tree defined by ξ conditioned on having exactly n vertices. One of the reasons for the interest in these trees is that many kinds of random trees occuring in various applications (random ordered trees, unordered trees, binary trees,...) can be seen as simply generated random trees and conditioned Galton–Watson trees, see e.g. Aldous [3, 4], Devroye [9] and Drmota [10]. It is easily seen that if a, b> 0 and we change wk to ˜wk: = ab k wk, (1.2)