Lower bound for d-sparse matrix multiplication in MPC

Establish a lower bound on the number of communication rounds required to multiply two d-sparse n×n matrices over semirings in the MPC model when there are n processors and each processor has O(d) local memory, under the standard assumption that at most d output entries per row and per column of the product are required.

Background

The paper studies matrix multiplication in the MPC model across several regimes and provides tight upper and lower bounds for dense and rectangular cases. For the sparse case, the authors consider n processors with O(d) memory per processor and assume interest in at most d output entries per row and column.

They present an O(d^0.9) upper bound for multiplying two d-sparse n×n matrices in semirings, improving upon a trivial O(d) bound. However, they explicitly state that they could not prove a lower bound for this setting, and note that some instances admit O(sqrt(d)) round algorithms. This leaves the round complexity without a proven lower bound in the stated sparse regime.