##### ###### ##### ### # # ### # # ###### ## ## ## ## ## ## ## # # # # # ## ##### #### ##### # # # # # # # #### ## # ## ## ## ## # # # # # ## ## # ###### ## ### # ### # ######
##### ###### ##### ### # # ### # # ###### ## ## ## ## ## ## ## # # # # # ## ##### #### ##### # # # # # # # #### ## # ## ## ## ## # # # # # ## ## # ###### ## ### # ### # ######
Core idea
When cost satisfies QI, optimal splits are monotonic. This lets you prune the search space.
Identify the DP type: Layered (partitioning) → D&C. Interval (combining) → Knuth.
Prove QI: .
Apply the improvement: Use divide-and-conquer or double bounds.
Track opt values: They bound future computations. This improvement reduces redundant work largely.
