I've been working on a caching trick that cuts down the runtime of a recursive search from exponential to linear in practice—mind if we dissect it together?

The trick is to store every sub‑problem result the first time you hit it, then just pull it back instead of recomputing. In code I use a dictionary keyed by the function’s arguments, and before the recursive call I check the cache. If the key is there I return that value, otherwise I compute, store, and return. It turns a naïve exponential recursion into essentially linear time for the distinct inputs.

Yeah, I ran it on a depth‑first maze‑solver where each cell can lead to up to four neighbours. I kept a tuple of the current position and the set of visited cells as the key. The branching was brutal—hundreds of thousands of paths—but after the first sweep the cache turned the remaining recursive calls into constant‑time lookups, so the total runtime dropped from minutes to a couple of seconds. The key was hashing the visited set efficiently with a frozenset so the key stayed immutable.