How can I optimize a dynamic programming algorithm better than O(n²)?

Achieving O(n log n) could be possible for specific dynamic programming problems, especially with sorting or searching in matrices, but it can get complex. The Toom-Cook multiplication algorithm is a practical method that might help in some cases. Just keep in mind it could be quite intricate to code.

Have you looked into using a monotonic deque? You might be able to traverse the grid linearly while managing the mins and maxes to find the optimal max-min difference. This could potentially lead you to an O(n) time complexity with O(k) space, which might fit your needs.

Since you’re dealing with a DP problem, practicing on platforms like LeetCode can really help you get a feel for the concepts. It’s definitely a good idea to familiarize yourself with common patterns in dynamic programming; it really is half the battle!

I’ve successfully used the Transfer Matrix method for similar problems before! It could be worth checking out. It’s a robust approach for optimization tasks as well. You can find more about it on Wikipedia if you're interested!

It’s tricky to give advice without full details, but typically, dynamic programming works by precomputing answers to avoid repeating calculations. Since you have to read the entire grid, getting below O(m*n) seems pretty challenging unless you find a way to skip certain candidates altogether. Perhaps consider if the problem fits a greedy or graph-based approach instead?