AI 中文总结
本文针对k²树传统层级式布局缓存性能差的问题，提出四种深度优先表示及线性时间压缩方法，实验表明其在矩阵操作中压缩与计算性能优于传统布局。
AI 中文摘要
本文研究面向图的静态、易于计算的无损压缩格式，聚焦于k²树的内存局部性与操作效率。我们发现k²树传统的层级式布局因局部性弱导致缓存性能差，尤其在矩阵-向量、矩阵-矩阵等操作中表现突出。为解决该局限，我们提出k²树的四种深度优先表示：普通深度优先布局（EDF-1）、平衡括号表示（BP）及其压缩变体（CEDF、CBP）；还引入基于后缀数组与最长公共前缀（LCP）数组的线性时间压缩方法，以识别并压缩相同子树。我们在两个真实数据集（Web Graphs、Wikidata）与一个合成数据集（随机邻接矩阵）上，针对上述线性代数操作，对比经典层级式k²树与基于DFUDS的表示，评估所提方法的执行时间、磁盘空间及峰值内存使用。结果显示，所提深度优先布局具竞争力且常优于现有方法：CEDF在多数场景实现最优压缩，EDF-1与CEDF持续降低峰值内存，性能随工作负载变化，不同布局在不同操作与数据场景中表现更佳。总体而言，本研究表明k²树的深度优先布局是传统布局的实用高效替代方案，可提升矩阵操作的压缩性能与计算性能。
英文摘要
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.
Comments44 pages, 7 figures, 18 tables
