Interview Questions: Minimum Spanning Trees

Minimum Spanning Trees

Bottleneck minimum spanning tree. Given a connected edge-weighted graph, design an efficient algorithm to find a minimum bottleneck spanning tree. The bottleneck capacity of a spanning tree is the weights of its largest edge. A minimum bottleneck spanning tree is a spanning tree of minimum bottleneck capacity.

Hint: prove that an MST is a minimum bottleneck spanning tree.

Extra challenge: Compute a minimum bottleneck spanning tree in linear time in the worst case. Assume that you can compute the median of n keys in linear time in the worst case.

反证法证明最小生成树一定是瓶颈生成树：假如最小生成树不是瓶颈生成树，记最小生成树$T$的最大权边为e，则存在一棵瓶颈树$T_b$，$T_b$中的所有边的权值都小于e的权值。删除 T T