I have a complete weighted graph G(V, E). I want to partition V into two clusters such that maximum intra-cluster edge length gets minimized. What is the fastest algorithm that solves this problem? I believe this can be solved in O(n^2) time where |V|=n. One approach would be making the graph bipartite. I could not figure out the complete algorithm. Can anyone help me to figure out the complete algorithm?
Two-color (depth-first search, O(n) time) a maximum spanning forest (Prim's algorithm, O(n2) time). Proof of correctness left as an exercise.
For the record, for sparser graphs with only m edges, I'm pretty sure there's an O(m)-time algorithm.