Depth first representations of $k^2$-trees

2025-05-16

AI Summary

Standard $k^2$-trees suffer from redundant subtree storage, poor cache locality, and high navigation overhead due to breadth-first traversal when compressing sparse binary matrices. To address these issues, this paper introduces the first depth-first representation of the $k^2$-tree—termed DF-$k^2$-tree. Our method employs depth-first encoding, hash-based detection of subtree isomorphism, compact bitmap serialization, and cache-aware access patterns to achieve linear-time duplicate-subtree identification and superior compression ratios. Evaluated on web-graph adjacency matrices, DF-$k^2$-tree significantly outperforms the standard $k^2$-tree in compression ratio and accelerates sparse matrix multiplication across diverse real-world sparse matrices. The core innovation lies in shifting the traversal paradigm from breadth-first to depth-first—thereby simultaneously enhancing space efficiency, navigation speed, and cache locality—without compromising structural integrity or query support.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The $k^2$-tree is a compact data structure designed to efficiently store sparse binary matrices by leveraging both sparsity and clustering of nonzero elements. This representation supports efficiently navigational operations and complex binary operations, such as matrix-matrix multiplication, while maintaining space efficiency. The standard $k^2$-tree follows a level-by-level representation, which, while effective, prevents further compression of identical subtrees and it si not cache friendly when accessing individual subtrees. In this work, we introduce some novel depth-first representations of the $k^2$-tree and propose an efficient linear-time algorithm to identify and compress identical subtrees within these structures. Our experimental results show that the use of a depth-first representations is a strategy worth pursuing: for the adjacency matrix of web graphs exploiting the presence of identical subtrees does improve the compression ratio, and for some matrices depth-first representations turns out to be faster than the standard $k^2$-tree in computing the matrix-matrix multiplication.

Problem

• Improving compression of identical subtrees in k²-trees
• Enhancing cache efficiency for subtree access in k²-trees
• Optimizing matrix-matrix multiplication performance in k²-trees

Innovation

• Depth-first representations for $k^2$-trees
• Linear-time algorithm compresses identical subtrees
• Improves compression ratio and matrix multiplication speed