Parallel Joinable B-Trees in the Fork-Join I/O Model

TL;DR

This paper introduces a new Fork-Join I/O Model to analyze and optimize I/O costs in parallel set operations on B-trees, achieving provable efficiency bounds for union, intersection, and difference operations.

Contribution

It proposes a novel I/O-efficient parallel algorithm for B-tree set operations within a new computational model, providing tight bounds on I/O costs.

Findings

Achieves $O(m \, \log_B(n/m))$ I/O work for set operations.

Provides bounds on I/O span: $O(\log_B m \cdot \log_2 \log_B n + \log_B n)$.

Introduces the Fork-Join I/O Model for analyzing parallel I/O costs.

Abstract

Balanced search trees are widely used in computer science to efficiently maintain dynamic ordered data. To support efficient set operations (e.g., union, intersection, difference) using trees, the join-based framework is widely studied. This framework has received particular attention in the parallel setting, and has been shown to be effective in enabling simple and theoretically efficient set operations on trees. Despite the widespread adoption of parallel join-based trees, a major drawback of previous work on such data structures is the inefficiency of their input/output (I/O) access patterns. Some recent work (e.g., C-trees and PaC-trees) focused on more I/O-friendly implementations of these algorithms. Surprisingly, however, there have been no results on bounding the I/O-costs for these algorithms. It remains open whether these algorithms can provide tight, provable guarantees in I/O-costs on trees. This paper studies efficient parallel algorithms for set operations based on search tree algorithms using a join-based framework, with a special focus on achieving I/O efficiency in these algorithms. To better capture the I/O-efficiency in these algorithms in parallel, we introduce a new computational model, Fork-Join I/O Model, to measure the I/O costs in fork-join parallelism. This model measures the total block transfers (I/O work) and their critical path (I/O span). Under this model, we propose our new solution based on B-trees. Our parallel algorithm computes the union, intersection, and difference of two B-trees with $O(m \log_B(n/m))$ I/O work and $O(\log_B m \cdot \log_2 \log_B n + \log_B n)$ I/O span, where $n$ and $m \leq n$ are the sizes of the two trees, and $B$ is the block size.