Worst-Case Optimal Tree Layout in External Memory

TL;DR

This paper establishes the theoretical limits for optimally laying out tree data structures in external memory to minimize worst-case block transfers during root-to-node traversals, across various depth regimes.

Contribution

It provides tight bounds on the minimum number of block transfers needed for tree traversals in external memory, filling a gap in understanding optimal tree layouts.

Findings

Optimal transfer bounds depend on tree depth and block size.

Different regimes of depth D yield different asymptotic transfer complexities.

The results guide efficient external memory tree data structure design.

Abstract

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 $$ \cases{ \displaystyle \Theta\left( {D \over \lg (1{+}B)} \right) & when $D = O(\lg N)$, \cr \displaystyle \Theta\left( {\lg N \over \lg \left(1{+}{B \lg N \over D}\right)} \right) & when $D = \Omega(\lg N)$ and $D = O(B \lg N)$, \cr \displaystyle \Theta\left( {D \over B} \right) & when $D = \Omega(B \lg N)$. } $$