Past exam of the computer science course of the University of Oxford / 2022 / Artificial intelligence / 2 / c / Solution

Let $D(p,q)=|p_x-q_x|+|p_y-q_y|$ be their Manhattan distance. Each joint action can reduce $D$ by at most two, so use the simultaneous rendezvous distanceThis heuristic function is nonzero whenever the robots are apart. For a transition $(p,q)\to (p^{\prime },q^{\prime })$, the triangle inequality givesApplying the ceiling function yields $h(p,q)\le 1+h(p^{\prime },q^{\prime })$, and $h(p,p)=0$. Thus $h$ is a consistent heuristic, hence an admissible heuristic.

In the unobstructed grid this estimate is exact. Along a shortest route of length $D$, both robots approach a middle room, taking at most $\lceil D/2\rceil$ steps each; when $D$ is an odd integer, the robot with the shorter route waits once. That final wait prevents a crossing from being mistaken for a meeting. With obstacles the same formula remains a relaxation heuristic, though it need not be exact.