Worst-Case Optimal Tree Layout in External Memory

Consider laying out a fixed-topology binary tree of N nodes into external memory with block size B so as to minimize the worst-case number of block memory transfers required to traverse a path from the root to a node of depth D. We prove that the optimal number of memory transfers is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{cases} \varTheta( {D \over\lg(1{+}B)} ) & \mathrm{when}~D = O(\lg N), \\ \varTheta( {\lg N \over\lg(1{+}{B \lg N \over D} )} ) & \mathrm{when}~D = \varOmega(\lg N)~\mathrm{and}~D = O(B \lg N), \\ \varTheta( {D \over B} ) & \mathrm{when}~D = \varOmega(B \lg N). \end{cases} \end{aligned}$$ \end{document}