# Understanding the proof that A* search is optimal

I don't understand the proof that $$A^*$$ is optimal.

The proof is by contradiction:

Assume $$A^*$$ returns $$p$$ but there exists a $$p'$$ that is cheaper. When $$p$$ is chosen from the frontier, assume $$p''$$ (Which is part of the path $$p'$$) is chosen from the frontier. Since $$p$$ was chosen before $$p''$$, then we have $$\text{cost}(p) + \text{heuristic}(p) \leq \text{cost}(p'') + \text{heuristic}(p'')$$. $$p$$ ends at goal, therefore the $$\text{heuristic}(p) = 0$$. Therefore $$\text{cost}(p) \leq \text{cost}(p'') + \text{heuristic}(p'') \leq \text{cost}(p')$$ because heuristics are admissible. Therefore we have a contradiction.

I am confused: can't we also assume there's a cheaper path that's in a frontier closer to the start node than $$p$$? Or is part of the proof that's not possible because $$A^*$$ would have examined that path because it is like BFS with lowest cost search, so, if there's a cheaper path, it'll be at a further frontier?

## 1 Answer

The key phrase here is

because heuristics are admissible

In other words, the heuristics never overestimate the path length:

$$cost(n) + heuristic(n) \le cost(\text{any path going through n})$$

And since the frontier is ordered by $$\textbf{cost + heuristic}$$, when a completed path $$p$$ is dequeued from the frontier, we know that it must necessarily be $$\le$$ any path going through some other frontier node $$q$$, because

$$cost(p) = cost(p) + heuristic(p) \le cost(q) + heuristic(q) \le cost(\text{any path going through q})$$

• However, $p'$ in the proof is not any path, it is a path assumed to be cheaper than $p$. So, how do you explain the conclusion "Therefore $\text{cost}(p) \leq \text{cost}(p'') + \text{heuristic}(p'') \leq \text{cost}(p')$"? By assumption, $\text{cost}(p') < \text{cost}(p)$. How come that you can conclude the opposite only because you say, in the proof, that you remove $p$ before $p''$ from the frontier? Even if the frontier is ordered, it does not imply that you have not removed $p'$ before $p$. I think this is what is confusing in the provided proof. – nbro Jun 27 '19 at 18:22
• @nbro: It is a proof by contradiction. That is the contradiction. You can in fact assume that $p'$ has not been dequeued yet, because if it had, the algorithm would have terminated and returned $p'$ – BlueRaja - Danny Pflughoeft Jun 27 '19 at 18:32
• I think that's the actual answer to the original question "can't we also assume there's a cheaper path that's in a frontier closer to the start node than $p$?". – nbro Jun 27 '19 at 18:36
• I mean, the answer to that question is really the proof given by OP. My answer is simply an alternative proof that more directly answers his question, hopefully in a way that makes more intuitive sense. Neither of these are the same as your original question, which should technically be a separate question on this site, but is easily answered in a comment. – BlueRaja - Danny Pflughoeft Jun 27 '19 at 18:47
• How is my question different than "can't we also assume there's a cheaper path that's in a frontier closer to the start node than 𝑝?", which, I suppose, it means "can't we also assume that $p'$ is dequeued before $p$"? The proof simply tells you, in an intricate way, that you remove $p$ before $p'$, hence $p'$ cannot be cheaper. By the way, how is yours an alternative proof? You just rephrased parts of the proof. – nbro Jun 27 '19 at 18:49