Interval DP: Mental Model

Start with the idea

Begin with the concrete question: Return maximum coins from bursting balloons. For the visible input [3,1,5,8], the stated result is 167. Before reading code, name the smallest piece of information that must survive from one step to the next. A multidimensional or structured DP state records the information needed to identify a subproblem: two prefixes, an interval, a bitmask, a tree node, or a resource limit. That is the mental model for Interval DP; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Solve increasing interval lengths and choose the last or first action inside each interval. Treat it as a claim to test after every operation. Write down the state before the first operation, update it once, and ask which part of the claim is still true. If an item is skipped, inserted, or removed, explain why that action cannot destroy an answer that should be retained. A useful trace has columns for the current input position, the state before the step, the action, and the state afterward. Use the visible sample until the expected result becomes inevitable rather than merely plausible.

Now inspect the Rust sketch, not just its final expression. The supplied Rust sketch is a starting point.