Computer Engineering(CE)

Question 1: Algorithms

August 2013

Problem 3. The minimum bottle neck spanning tree (MBST) is a spanning tree that seeks to minimize the use of most expensive (the largest weight) edge in the tree. More specifically, for a tree $T$ over a graph G, we define $e$ to be a bottleneck edge of $T$ if it's an edge with the largest weight. Note, multiple edges may have the same weight. The tree $T$ is an MBST if it is a spanning tree and there is no other spanning tree of G with a cheaper bottleneck edge. Prove or disprove that an MBST for a graph G is always a minimum spanning tree for G.

### Solution 1

A minimum bottle neck spanning tree (MBST) is not always a minimum spanning tree MST.

Proof: Suppose we have a spanning tree $T$ which is a MBST of $G$ , and $T$ is also a MST. The bottle neck is edge $e$ with weight $w$. Then I have another vertex $X$ that will be connected to the graph $G$ to form a new graphs $G'$. $G' = G + X$. And the possible edges that connect graph $G'$ and new vertex $X$ has weights $w_1, w_2, ... W_x$, and each is no greater than the bottle neck weight $w$. Using any of the edges from $w_1, w_2, ... W_x$, I get a new tree $T'$. So $T'$ is still a spanning tree, and its largest weight remains $w$, so $T'$ is also an MBST. However, among these edges that I can choose to span the tree, only the one that has the smallest weight will be a MST, by definition. So an MBST for a graph $G'$ is not a minimum spanning tree for $G'$. End of proof.

### Solution 2

A MBST tree is not a minimum spanning tree for $G$, ABST may not contain the lightest edge. A counter example is showing in figure below: