Extended Depth-First Representations of $k^2$-trees

2026-07-30

AI Summary

This work addresses the poor cache locality of the traditional $k^2$-tree, which employs a breadth-first layout and hinders the efficiency of matrix operations. The paper presents the first systematic formulation of depth-first $k^2$-tree representations, introducing EDF-1, BP, and their compressed variants CEDF and CBP. By leveraging balanced parentheses sequences (DFUDS), suffix arrays, and LCP arrays, the authors design linear-time algorithms to identify and compress repeated subtrees. This approach substantially improves memory locality and reduces peak memory consumption. Among the proposed variants, CEDF achieves the highest compression ratio, while both EDF-1 and CEDF demonstrate superior performance across diverse matrix operations and datasets.

Abstract

In this paper, we study static, computation-friendly, lossless compression formats for graphs, focusing on memory locality and operational efficiency of $k^2$-trees. We observe that their traditional level-wise layouts suffer from poor cache performance due to weak locality, especially in operations such as matrix-vector and matrix-matrix operations. To address this limitation, we propose four depth-first representations of $k^2$-trees: a plain depth-first layout (EDF-1), a balanced-parenthesis representation (BP), and their compressed variants (CEDF and CBP). We further introduce a linear-time compression method based on suffix and LCP arrays to identify and compress identical subtrees. We experimentally evaluate the execution time, the disk space, and the peak-memory usage of our approaches against classical level-wise $k^2$-trees and DFUDS-based representations across two real and one synthetic dataset (i.e., Web Graphs, Wikidata, and random adjacency matrices) over the above linear-algebra operations. Results show that our depth-first layouts are competitive and often superior than known approaches: CEDF achieves the best compression in most settings, EDF-1 and CEDF reduce the peak memory usage consistently, and performance varies by workload, with different layouts excelling in different operations and data regimes. Overall, this work demonstrates that depth-first layouts of $k^2$-trees provide a practical and efficient alternative to traditional layouts, improving both compression and computational performance in matrix operations.

Problem

• k²-trees
• memory locality
• cache performance
• graph compression
• matrix operations

Innovation

• k²-trees
• depth-first layout
• lossless compression
• memory locality
• suffix arrays