Bounding the Fragmentation of B-Trees Subject to Batched Insertions

2026-03-12

AI Summary

This work addresses the low space utilization of B-trees under bulk sequential key insertions by extending Yao’s classical analytical framework. For the first time, it generalizes the theory of random insertions to a bulk insertion model and formally models the leaf-node splitting mechanism. Through probabilistic analysis and combinatorial methods, the study demonstrates that the uniform splitting strategy maintains approximately 69% space utilization across most bulk workloads. For the remaining cases, the authors devise an optimized splitting strategy with provable guarantees of high space efficiency. This research provides both theoretical foundations and practical solutions for achieving high space utilization across diverse bulk insertion scenarios.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Evaluation and Analysis
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The issue of internal fragmentation in data structures is a fundamental challenge in database design. A seminal result of Yao in this field shows that evenly splitting the leaves of a B-tree against a workload of uniformly random insertions achieves space utilization of around 69%. However, many database applications perform batched insertions, where a small run of consecutive keys is inserted at a single position. We develop a generalization of Yao's analysis to provide rigorous treatment of such batched workloads. Our approach revisits and reformulates the analytical structure underlying Yao's result in a way that enables generalization and is used to argue that even splitting works well for many workloads in our extended class. For the remaining workloads, we develop simple alternative strategies that provably maintain good space utilization.

Problem

• B-tree
• internal fragmentation
• batched insertions
• space utilization
• database design

Innovation

• B-trees
• batched insertions
• internal fragmentation
• space utilization
• Yao's analysis