I have a fully connected graph $G=(V,E)$ with $n$ vertices. The edge weights $w(e)$ with $e\in E$ are non-negative and form a metric space (e.g. Hamming distance), thus for vertices $v,u,y \in V$, we have $w(v,y) \leq w(v,u)+w(u,y)$.
However, it is expensive to calculate $w(\cdot)$.
My question is, is there an algorithm that can calculate the minimum spanning tree, without calculating all $n(n-1)/2$ edge weights?