Assume we have a set of points in the euclidean space. We can easily find the 2-center clustering of those points. Basically we have to cover the points in the plane by two circular disks, in such a way as to minimize the radius of the larger of the two disks. Each disk represents a cluster. Our main focus is to find the clusters, not the centers. There are existing geometric algorithms for this problem.
However, assume we only have the pairwise distance matrix of the points (no coordinates). We can think of it as a complete graph where an edge weight is the Euclidean distance between two points. Can we find the exact 2-center clustering of the point set from this complete graph? We can apply the graph-based 2-center algorithms. Is it equivalent to the actual geometric 2-center clustering? Is there any example where these two cases produce different clustering?