Algorithmica

, Volume 72, Issue 2, pp 369–378 | Cite as

Worst-Case Optimal Tree Layout in External Memory

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Abstract

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
$$\begin{aligned} \begin{cases} \varTheta( {D \over\lg(1{+}B)} ) & \mathrm{when}~D = O(\lg N), \\ \varTheta( {\lg N \over\lg(1{+}{B \lg N \over D} )} ) & \mathrm{when}~D = \varOmega(\lg N)~\mathrm{and}~D = O(B \lg N), \\ \varTheta( {D \over B} ) & \mathrm{when}~D = \varOmega(B \lg N). \end{cases} \end{aligned}$$

Keywords

Data structures Trees External-memory 

Notes

Acknowledgements

This research was initiated during the Seminar on Cache-Oblivious and Cache-Aware Algorithms held at Schloss Dagstuhl in July 2004. We thank Jeff Erickson and J. Ian Munro for many helpful discussions during that seminar. We also thank Paulo Ferragina for early discussions on this problem.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • John Iacono
    • 2
    • 3
  • Stefan Langerman
    • 4
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Polytechnic Institute of New York University (Formerly Polytechnic University)BrooklynUSA
  3. 3.MADALGO—Center for Massive Data Algorithmics, a Center of the Danish National Research FoundationAarhus UniversityAarhus NDenmark
  4. 4.Département d’informatique, Université Libre de BruxellesF.R.S.-FNRSBrusselsBelgium

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