Complexity of thin matrix-product output extraction

Characterize the true complexity of computing a prescribed set W of entries of the thin matrix product XY as a function of the inner dimension D and the number of requested positions |W|, including parameter regimes beyond D=N^{0.12}.

Background

The central algorithm computes sparse sets of entries of an N-by-D matrix multiplied by a D-by-N matrix, with strong savings when D is a sufficiently small power of N. The authors state that the full complexity of this output-extraction task is not known, particularly for larger inner dimensions, and suggest seeking new algebraic identities or related techniques.

References

We do not know the true complexity of computing a set $W$ of entries of a thin product $XY$, as a function of $D$ and $|W|$.

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 8, Section “Conclusion,” paragraph “Algebraic algorithm design”