On the variance of a class of inductive valuations of data structures for digital search

Abstract

Let an inductive valuation L on the family of binary tries or Patricia tries or digital search trees be defined in the following way: L(t) = L(tl) + L(tr) + R(t), where tl and tr denote the left and right subtrees of t and R depends only on the size (the number of records) ¦t¦ of t. Let LN denote L restricted to the trees of size N. In Theorem 1 we give sufficient conditions on the sequence r¦t¦$̈= R(t) for the variance Var LN to be of exact order N, if the family of tries (resp. Patricia tries, resp. digital search trees) is equipped with the Bernoulli model. For the symmetric Bernoulli model we prove the existence of a continuous periodic function δ with period 1, such that Var LN ∼ δ(log2 N) .̄ N holds.