A* Search: Mental Model

Start with the idea

Begin with the concrete question: On a 4x4 empty grid, return shortest moves from (0,0) to (3,2) using Manhattan heuristic. For the visible input start=0,0 goal=3,2, the stated result is 5. Before reading code, name the smallest piece of information that must survive from one step to the next. Edge weights change what a path means. State whether the objective is minimum distance, a spanning structure, or feasible flow before choosing state. That is the mental model for A* Search; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Prioritize distance-so-far plus an admissible estimate that never overstates remaining cost. 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 an invariant note rather than a complete program.