Minimum Spanning Tree

A minimum spanning tree is

• a tree, meaning it has no cycle
• minimum, meaning it has minimum weight
• spanning, meaning it connects all nodes

The MST problem takes a connected graph $G$ and outputs an MST for that graph.

Algorithm that solves this problem includes Kruskal's Algorithm and Prims Algorithm.

Suboptimality

The MST problem exhibits the optimal substructure property.

Given MST tree for the graph G

$OPT={\text{set of edges}}$

where the number of edges is $n-1$, the weight of edges is minimum, and the tree is spanning.

Consider $(x,y)$, where X is a leaf node.

Prove that $A=OPT-(x,y)$ is a MST for the subproblem $G-X$

By contradiction, assume that $A$ is not optimal. There must be $B$ such that

$w(B)<w(A)$

Adding the edge (x, y) back to both graphs

$B+(x,y)$ is a viable tree.

$w(B+w(x,y))<w(A+w(x,y))=w(OPT)$

Therefore, OPT is not the optimal solution.

By contradiction, $A$ must be optimal.