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Minimum Spanning Tree
We are interested in finding a minimum spanning tree of a connected, undirected graph. A minimum spanning tree is a subgraph of the original graph that is a tree and connects all the vertices together with the minimum possible total edge weight. Given a connected, undirected graph \(\mathcal{G} = (\mathcal{V},\mathcal{E})\), where \(\mathcal{V}\) is the set of vertices and \(\mathcal{E}\) denotes the set of edges. Each edge \(e \in \mathcal{E}\) has a weight \(w(e)\) associated with it.
Spanning Tree
A spanning tree contains a subset of edges \(\mathcal{E}'\subseteq\mathcal{E}\) with the following properties:
- The subset contains exactly \(n-1\) edges, where \(n\) is the number of vertices in the graph.
- The subset \(\mathcal{E}'\) connects all the vertices in the graph.
We seek to find the particular spanning tree that has the minimum total edge weight. This tree is called the minimum spanning tree. With this, we can formally define the problem as:
\begin{equation} MST = \arg\min_{\mathcal{E}’\subseteq \mathcal{E}} \sum w(e) \end{equation}
There are two most popular algorithms to construct a minimum spanning tree:
- Kruskal’s algorithm
- Prim’s algorithm