# Belga B-trees

## TL;DR

This paper introduces the Belga B-tree, a self-adjusting external memory data structure that adapts to query distributions and achieves near-optimal performance, extending ideas from binary search trees to B-trees.

## Contribution

The paper formalizes the B-Tree model, proves lower bounds, and presents the Belga B-tree, which is competitive with the best offline B-tree algorithms within a logarithmic factor.

## Key findings

- Belga B-tree achieves $O(\log \log N)$ competitiveness.
- Transformation from static BST to B-tree is faster by a $ heta(\log B)$ factor.
- Randomization is necessary for significant speedup in the transformation.

## Abstract

We revisit self-adjusting external 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. 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 $\Theta(\log B)$ factor; the transformation is randomized and we show that randomization is necessary to obtain any significant speedup.