Boolean Matrix Multiplication for Highly Clustered Data on the Congested Clique (2405.16103v1)

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.