Optimal Deterministic Fully Sparse Matrix Multiplication

Abstract

We give the first deterministic algorithm for fully sparse matrix multiplication that attains the optimal running-time exponent. This result matches the best previously known randomized algorithm running-time exponent. Given compatible matrices $A$ and $B$ over an arbitrary associative ring with identity, with $\operatorname{nnz}(A),\operatorname{nnz}(B)=O(n^{\delta_{\mathrm{in}}})$ and $\operatorname{nnz}(AB)=O(n^{\delta_{\mathrm{out}}})$, our algorithm finds the support of $AB$ and computes the product exactly in $$O\!\left(n^{\beta_R(\delta_{\mathrm{in}},\min\{\delta_{\mathrm{out}},2\delta_{\mathrm{in}}\})+\varepsilon}\right)$$ operations, where $\beta_R(\delta_{\mathrm{in}},\delta)$ denotes the maximum of $\delta_{\mathrm{in}}$ and $\omega_{\delta_{\mathrm{in}},R}(a,1,b)$ over all $a,b\in[0,1]$ satisfying $a+b=\delta$. For dense inputs over a commutative ring, this bound simplifies to $O(n^{\omega_R((\delta_{\mathrm{out}}-1)_+,1,1)+\varepsilon})$. With the current rectangular matrix multiplication bounds, this is nearly quadratic, namely $O(n^{2+\varepsilon})$, for every $\delta_\mathrm{out}\le1.321334$, improving the previous deterministic range of $\delta_{\mathrm{out}}\le 0.642668$. To prove this result, we develop a general deterministic recovery technique that finds and fixes sparse parts of an unknown matrix while keeping temporary errors in denser parts under control.