Deletion algorithms for binary search trees

Abstract

The effect of updating (deletions/insertions) on binary search trees has been an interesting research topic for almost three decades, but in the last five years there have been a few contributions, due partially to the intrinsic difficulty of the involved analysis. Since the problem is quite difficult to be solved in a general fashion, we have restricted ourselves to solve a simpler problem, which shall be considered as an important and necessary basis for further developments. In this paper, we have faced the systematization of the study of deletion algorithms, and deduced the effect that a single random deletion produces in the probability distribution of binary trees of arbitrary size for a wide variety of deletion algorithms. Furthermore, to carry on the analysis, some new tools have been introduced, such as the concepts of strong and weak invariance of probability functions induced by an algorithm. Among others, we have been able to derive interesting results such as an extension of Hibbard's classical theorem and sufficient conditions under which a complexity measure of main practical importance, the expected search time, does not change.