Limiting distributions for the costs of partial match retrievals in multidimensional tries

Abstract

In this paper we derive limiting distributions for the costs of partial match retrievals in K-d tries. We confine ourselves to the binary case (i.e., the alphabet, from which the letters of the keys are drawn, has cardinality 2), to dimension K=2, and to the query pattern (*, S). The probabilistic model that we assume is the Bernoulli model: keys are sequences of i.i.d. random variables, which assume the values 0 and 1 with probability p and 1−p, and are pairwise independent. In the symmetric case (p=½) our analysis reveals convergence in distribution of properly normalized costs CN, as N→∞ (N is the number of keys), to the normal distribution, as well as convergence of the moments. We also consider two models dealing with flexible queries. The first, which we call “adapted,” assumes that the decision, in which subtree of a node to continue the search, is made when we reach that node, independent of all other such decisions. The second model, which we call “predictable,” assumes that that decision is made one step in advance, telling, if the search should be continued in both left (resp. both right) subtrees of the two sons, again independent of all other such decisions. In the symmetric case the limiting distributions are the same as before, and in the asymmetric case limiting distributions are given in terms of some nonnormal distribution μp, which is the unique solution of a certain stochastic fixed point equation. Again we also establish convergence of the moments. Furthermore we derive some properties of μp from its fixed point equation, such as existence and infinite differentiability of the density, and analyticity of the moment generating function, and we also determine the support of μp. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 428–459, 2000