Find all edges of G contained in some MSP

Question

I have the following question in a homework:

Let $G = (V,E)$ be an undirected graph, and let $$A = \{ e \in E \mid \text{ s.t. exists an MSP $T$ containing } e\}.$$

We were asked to find $A$ in $O(m \log n)$ time.

Any suggestions?

Answer 1

We call an edge superheavy if it is the unique heaviest edge in some cycle. This post shows that an edge $e\in A$ if and only if $e$ is not superheavy.

To find all edges that are not superheavy, we can first sort the edges according to their weights from small to large. Then, we remove all edges, re-add them in this order, and track the connected components by a disjoint-set data structure.

Now we can check whether an edge $e$ is superheavy by checking whether it connects two vertices in different connected components in the graph with edges whose weights are less than $w(e)$: If $e$ is superheavy, then the two endpoints of $e$ are connected by a path on which the edges have weights less than $w(e)$, so the two endpoints must lie in the same connected components in the graph with edges whose weights are less than $w(e)$. On the other hand, if $e$ connects two vertices in the same connected components in the graph with edges whose weights are less than $w(e)$, then we can find a path connecting the two endpoints of $e$ on which the edges have weights less than $w(e)$, so $e$ along with this path forms a cycle and $e$ has the largest weight in this cycle, so $e$ is superheavy.