The Runner-Up Spanning Tree — Solution & Editorial

Back to the Problem

A second-best spanning tree differs from the MST by exactly one edge. Build the MST, then for each MST edge try removing it and rebuilding the cheapest spanning tree without it; take the minimum. (The classic O(N²) method precomputes max path edges.)

Complexity: O(N · M log M)

Watch out: Excluding an MST edge may disconnect the graph — the rebuild must verify a full spanning tree still forms.