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What Does Dynamic Optimality Mean in External Memory?

Authors Michael A. Bender, Martín Farach-Colton, William Kuszmaul

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Author Details

Michael A. Bender
  • Stony Brook University, Stony Brook, NY, USA
Martín Farach-Colton
  • Rutgers University, New Brunswick, NJ, USA
William Kuszmaul
  • MIT, Cambridge, MA, USA

Funding

  • Bender, Michael A.: Supported in part by NSF grants CCF-2118832, CCF-2106827, CCF-1725543, CSR-1763680, CCF-1716252, and CNS-1938709.
  • Farach-Colton, Martín: Supported in part by NSF grants CSR-1938180, CCF-2106999, and CCF-2118620.
  • Kuszmaul, William: Funded by a Hertz Fellowship and an NSF GRFP Fellowship. Research was also partially sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Acknowledgements

Portions of this work were completed at the Second Hawaii Workshop on Parallel Algorithms and Data Structures. The authors would like to thank Nodari Sitchinava for organizing the workshop.

Cite AsGet BibTex

Michael A. Bender, Martín Farach-Colton, and William Kuszmaul. What Does Dynamic Optimality Mean in External Memory?. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.18

Abstract

A data structure A is said to be dynamically optimal over a class of data structures 𝒞 if A is constant-competitive with every data structure C ∈ 𝒞. Much of the research on binary search trees in the past forty years has focused on studying dynamic optimality over the class of binary search trees that are modified via rotations (and indeed, the question of whether splay trees are dynamically optimal has gained notoriety as the so-called dynamic-optimality conjecture). Recently, researchers have extended this to consider dynamic optimality over certain classes of external-memory search trees. In particular, Demaine, Iacono, Koumoutsos, and Langerman propose a class of external-memory trees that support a notion of tree rotations, and then give an elegant data structure, called the Belga B-tree, that is within an O(log log N)-factor of being dynamically optimal over this class. In this paper, we revisit the question of how dynamic optimality should be defined in external memory. A defining characteristic of external-memory data structures is that there is a stark asymmetry between queries and inserts/updates/deletes: by making the former slightly asymptotically slower, one can make the latter significantly asymptotically faster (even allowing for operations with sub-constant amortized I/Os). This asymmetry makes it so that rotation-based search trees are not optimal (or even close to optimal) in insert/update/delete-heavy external-memory workloads. To study dynamic optimality for such workloads, one must consider a different class of data structures. The natural class of data structures to consider are what we call buffered-propagation trees. Such trees can adapt dynamically to the locality properties of an input sequence in order to optimize the interactions between different inserts/updates/deletes and queries. We also present a new form of beyond-worst-case analysis that allows for us to formally study a continuum between static and dynamic optimality. Finally, we give a novel data structure, called the Jεllo Tree, that is statically optimal and that achieves dynamic optimality for a large natural class of inputs defined by our beyond-worst-case analysis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
Keywords
  • Dynamic optimality
  • external memory
  • buffer propagation
  • search trees

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