A Characterization of Digital Search Trees from the Average Complexity Viewpoint

This paper smdies the average complexity of digital search trees from the successful search point of view. The average value of the successful search is used to evaluate the search time for a given record, the number of comparisons to insert a record, etc. The average value, however, is rather a poor measure and the need for higher moments of the successful search is obvious. For example, the variance provides information on "how well is a digital tree balanced"; the third centralized moment is a measure of the skewness property of the distribution, etc. In this paper we concentrate on an open problem: how to evaluate aU moments of the successful search in an asymmetric multiway digital search tree. We prove that the m·th successful search S/I. where n is the number of stored records, satisfies lim E Sgal1nm n = Vhf, where .h I is the entropy of the alphabet. In particular, it is shown that the variance of SrI is varSrI =c Inn +0 (1) for asymmetric case, and varSn = 0 (1) for symmetric case (we also determine the constant). This gives a complete characterization of the digital search tree from the successful search view point