Boolean Matrix Multiplication for Highly Clustered Data on the Congested Clique

TL;DR

This work addresses the Boolean matrix multiplication problem on the congested clique when input data are highly clustered, proposing a protocol whose round complexity scales as $\tilde{O}(\sqrt{\frac{M}{n}+1})$, with $M$ the minimum of the MST costs of the relevant input rows/columns in the Hamming space. The approach hinges on computing an approximate MST in $\{0,1\}^n$ via randomized dimension reduction, yielding an $O(\log^3 n)$-round HMST protocol, and then organizing computation and communication along a balanced traversal of the MST to distribute subproblems. The core contributions are (i) an exact MST computable in $O(n^{1-2/\omega+\epsilon})$ rounds, (ii) an $O(1)$-approximate MST in $O(\log^3 n)$ rounds, and (iii) a Boolean matrix product protocol in $\tilde{O}(\sqrt{\frac{M}{n}+1})$ rounds that outperforms generic matrix-multiplication-based schemes when data are highly clustered. These results extend to the transpose product and to extended Hamming distance, with a detailed work analysis showing $\tilde{O}(n(n+M))$ total work.

Abstract

We present a protocol for the Boolean matrix product of two $n\times b$ Boolean matrices on the congested clique designed for the situation when the rows of the first matrix or the columns of the second matrix are highly clustered in the space $\{0,1\}^n.$ With high probability (w.h.p), it uses $\tilde{O}\left(\sqrt {\frac M n+1}\right)$ rounds on the congested clique with $n$ nodes, where $M$ is the minimum of the cost of a minimum spanning tree (MST) of the rows of the first input matrix and the cost of an MST of the columns of the second input matrix in the Hamming space $\{0,1\}^n.$ A key step in our protocol is the computation of an approximate minimum spanning tree of a set of $n$ points in the space $\{0,1\}^n$. We provide a protocol for this problem (of interest in its own rights) based on a known randomized technique of dimension reduction in Hamming spaces. W.h.p., it constructs an $O(1)$-factor approximation of an MST of $n$ points in the Hamming space $\{ 0,\ 1\}^n$ using $O(\log^3 n)$ rounds on the congested clique with $n$ nodes.

Theorems & Definitions (28)

• lemma 1
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• lemma 2
• proof
• theorem 1
• proof
• remark 1
• lemma 3
• proof
• lemma 4