space complexity of DFA intersection problem
Solving intersection Non-Emptiness for 2 DFA's:

It essentially just becomes a reachability problem for the product DFA.


Roughly, we can solve it deterministically in $O(n^2)$ time using $O(n^2)$ space.
Or, we can solve it non-deterministically with $O(\log(n))$ space.
By Savitch's Theorem, we can also solve it deterministically in $2^{O(\log^2(n))}$ time using $O(\log^2(n))$ space.


To the best of my knowledge, it is not known if we can solve it deterministically in polynomial time using $O(n^{2-\epsilon})$ space.

Feel free to share any thoughts or ideas in the comments.  :)