Given two sets $A, B\subset \mathbb{N}^2$, each with finite cardinality, what's the most efficient algorithm to compute $\min_{u\in A, v\in B}d(u, v)$ where $d(u,v)$ is the (Euclidean) distance between $u$ and $v$ (may be considered as points or vectors). Also what's the most efficient algorithm to determine $u$ and $v$ (not necessarily unique) such that $d(u,v)$ is minimized?
So minimizing $d(u,v)$ is pretty much the same as minimizing $[d(u,v)]^2$, which can be written using the distance formula. But I couldn't think of anything better than a brute-force $O(|A||B|)$ solution (where $|A|$ denotes the cardinality of $A$) which tries every possible $u\in A,v\in B$.