Dynamic k-dimensional multiway search under time-varying access frequencies

We consider multiway search trees for k-dimensional search under time-varying access frequencies. Let S = {kl,...,kn} be a set of k-dimensional keys, k≥1, and let pit be the number of accesses to ki, also called frequency of ki, up to time t, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$W^t = \sum\limits_{i = 1}^n {p_i^t }$$ \end{document}. We present weighted (k+1)B-trees of order d, d≥1, with the following properties:A search for key ki can be performed in time 0(min(n,logd+1Wt/pit)+(k−1)), i.e. the tree is always nearly optimal.The time for updating after a search is at most proportional to search time.Insertion of a new key with arbitrary frequency as well as deletion of a key with arbitrary frequency can be carried out in time 0(min(n,logd+1Wt)+(k−1)).