# Single source shortest path over non-commutative finite idempotent semiring in Cartesian product

Let $$G$$ be a Cartesian product of two arbitrary directed weighted graphs $$M$$ and $$N$$. The weights are from a non-commutative finite idempotent semiring.

Do there exist advanced results on the single source shortest path (SSSP) problem for $$G$$? Suppose that $$|V_M| = |V_N| = n$$. Is it possible to solve SSSP in $$\widetilde{O}(BMM(n))$$?