How can I optimize a dynamic programming algorithm from O(n²) to a better time complexity?

I've found success using the Transfer Matrix approach for similar problems. You can check out more about it here: https://en.wikipedia.org/wiki/Transfer-matrix_method_(statistical_mechanics). It can be a handy method, depending on the exact nature of your problem!

Just having it identified as a DP problem is a significant first step. You might want to tackle some practice problems on coding platforms to get a good handle on common techniques. Seeing the applications of DP firsthand could reveal new strategies you haven't considered yet.

It’s tough to say without knowing all the specifics, but typically, dynamic programming (DP) focuses on precomputing or memoizing results to cut down on repeated work. Given that you're handling a grid, achieving better than O(m*n) seems tricky since you usually need to inspect the entire grid as part of the process. You might need to look into greedy or graph-based approaches that could potentially eliminate candidates without having to evaluate the entire grid.

I appreciate the input! I'm intentionally holding back on sharing more specifics due to academic integrity. Just trying to find my way through the problem.

While I don't know all the constraints you're facing, consider using a monotonic deque strategy. You could process the grid in a linear fashion, adjusting the mins and maxes as you go. This might allow you to find the max-min difference in O(n) time with O(k) space, which sounds like a possible approach here!

When it comes to dynamic programming involving matrix operations, O(n log n) is a pretty solid theoretical limit for sorting or searching tasks. There are certain algorithms that can achieve better than O(n²) complexity, but they tend to be tricky to implement. You might want to explore Toom-Cook multiplication; it's generally simpler for practical applications.