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Belga B-Trees

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Abstract

We revisitself-adjustingexternal memory tree data structures, which combine the optimal (and practical) worst-case I/O performances of B-trees, while adapting to the online distribution of queries. Our approach is analogous to undergoing efforts in the BST model, where Tango Trees (Demaine et al., SIAM J. Comput. 37(1), 240–251, 2007) were shown to be \(O(\log \log N)\)-competitive with the runtime of the best offline binary search tree on every sequence of searches. Here we formalize the B-Tree model as a natural generalization of the BST model. We prove lower bounds for the B-Tree model, and introduce a B-Tree model data structure, the Belga B-tree, that executes any sequence of searches within a \(O(\log \log N)\) factor of the best offline B-tree model algorithm, provided \(B=\log ^{O(1)}N\). We also show how to transform any static BST into a static B-tree which is faster by a \({\varTheta }(\log B)\) factor; the transformation is randomized and we show that randomization is necessary to obtain any significant speedup.

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Notes

  1. The Tango tree was invented on an overnight flight from JFK airport en route to Buenos Aires, Argentina. The work on the Belga B-Tree has been substantially completed at Cafe Belga, Ixelles, Belgium.

  2. Recall that in the external memory model (defined in Section 1) B denotes the block size. Each B-tree node has at most B children, contains O(B) words and thus it can be stored in O(1) blocks of size B.

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Authors and Affiliations

  1. CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

    Erik D. Demaine

  2. Université libre de Bruxelles, Brussels, Belgium

    John Iacono, Grigorios Koumoutsos & Stefan Langerman

  3. New York University, New York, NY, USA

    John Iacono

  4. Fonds de la Recherche Scientifique (F.R.S.-FNRS), Brussels, Belgium

    Stefan Langerman

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  1. Erik D. Demaine
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  4. Stefan Langerman
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Correspondence to Grigorios Koumoutsos.

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This article belongs to the Topical Collection: Special Issue on Computer Science Symposium in Russia (2019)

Guest Editor: Gregory Kucherov

This work was supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001 1 and by NSF Grant CCF-1533564.A preliminary version of this article appeared in the Proceedings of the 14th International Computer Science Symposium in Russia (CSR 2019).

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Demaine, E.D., Iacono, J., Koumoutsos, G. et al. Belga B-Trees. Theory Comput Syst 65, 541–558 (2021). https://doi.org/10.1007/s00224-020-09991-8

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