Depth first representations of $k^2$-trees

TL;DR

This paper introduces depth-first representations of the $k^2$-tree, enhancing compression and efficiency for sparse binary matrices, especially in web graph adjacency matrices and matrix multiplication tasks.

Contribution

It proposes novel depth-first $k^2$-tree representations and an algorithm for compressing identical subtrees, improving compression and computational speed.

Findings

Depth-first representations improve compression ratios for web graph adjacency matrices.

Depth-first structures can outperform standard $k^2$-trees in matrix-matrix multiplication.

Experimental results validate the efficiency and compression benefits of the proposed approach.

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.