Top-Down Memoization: When to Use It

Start with the idea

A problem is a good match for Top-Down Memoization when it asks for recursive recurrences with repeated states. Here the concrete task is: Return Fibonacci number F(12) with F(0)=0 and F(1)=1. The visible input n=12 is an example, not a definition of the whole problem. Ask what operations the solution must support and what the input guarantees. Choose a recurrence when subproblems overlap. Some patterns summarize a frontier instead of a full DP table; either representation needs an invariant, and any greedy shortcut needs proof. The right choice follows from those guarantees, not from a familiar noun in the prompt.

The decisive invariant is: Cache a recursive state the first time it is solved so overlapping subproblems run once. Try to say why it applies to the task without quoting the implementation. Which fact becomes known after one update, and which possible answers remain? A one-dimensional DP state summarizes a prefix or a single parameter. Define its meaning before writing a recurrence. If that claim cannot be maintained on a legal input, the pattern is wrong or one of its preconditions is missing. A counterexample is more useful than a vague warning: change one property of the sample, trace the first two steps, and identify the exact claim that fails.

Compare costs before committing. The advertised bound is O(states * transition) time · O(states) space; identify the operation responsible for it and the extra memory it needs.