Worst-Case Optimal Tree Layout in External Memory

Consider laying out a fixed-topology 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 $$ \cases{ \displaystyle Θ\left( {D \over \lg (1{+}B)} \right) & when $D = O(\lg N)$, \cr \displaystyle Θ\left( {\lg N \over \lg \left(1{+}{B \lg N \over D}\right)} \right) & when $D = Ω(\lg N)$ and $D = O(B \lg N)$, \cr \displaystyle Θ\left( {D \over B} \right) & when $D = Ω(B \lg N)$. } $$

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