What Is MST (Minimum Spanning Tree)?

What is a Minimum Spanning Tree (MST)?

A Minimum Spanning Tree (MST) is a fundamental concept in graph theory, especially in the context of weighted graphs. In simple terms, an MST of a graph is a subgraph that connects all the vertices (nodes) of the original graph together with the minimum possible total edge weight, while ensuring that no cycles are formed.

In other words, it’s a tree that connects all the vertices of the graph with the least cost, where the “cost” is usually represented by the weights of the edges.

Key Characteristics of a Minimum Spanning Tree:

1. Spanning: It includes all the vertices from the original graph.
2. Tree: It has no cycles and is connected (i.e., there’s a path between any two vertices).
3. Minimum Weight: The sum of the weights of the edges in the tree is the minimum possible when compared to all other spanning trees of the graph.

Graph Terminology:

• Vertex (Plural: Vertices): A node or point in a graph.
• Edge: A connection between two vertices, which may have a weight or cost associated with it.
• Weighted Graph: A graph where each edge has an associated weight or cost.
• Tree: A connected graph with no cycles.
• Spanning Tree: A subgraph that contains all the vertices of the graph and is a tree.

Example:

Consider the following weighted undirected graph:

    A---2---B
   / \      /
  6   3    5
 /     \  /
C---4----D

In this graph:

• Vertices: A, B, C, D.
• Edges with weights: (A-B: 2), (A-C: 6), (A-D: 3), (B-D: 5), (C-D: 4).

The Minimum Spanning Tree (MST) is a subset of these edges that connects all the vertices with the least weight sum.

Algorithms for Finding the MST:

There are several well-known algorithms to find the Minimum Spanning Tree of a graph. Some of the most commonly used algorithms are:

1. Kruskal’s Algorithm:

Kruskal’s algorithm is a greedy algorithm that builds the MST by selecting the edges in increasing order of their weights, as long as they don’t form a cycle. Here’s how it works:

• Sort all the edges of the graph in ascending order of their weight.
• Pick the smallest edge that does not form a cycle with the edges already included in the MST.
• Repeat the process until all vertices are connected.
• Kruskal’s algorithm uses a disjoint-set (union-find) data structure to check whether two vertices are already connected to avoid cycles.

Steps of Kruskal’s Algorithm:

1. Sort all edges in non-decreasing order of weight.
2. Pick the smallest edge. If adding this edge doesn’t form a cycle, include it in the MST.
3. Repeat the process until the tree contains V−1V – 1 edges, where VV is the number of vertices in the graph.

Example (using the graph above):

• Sort the edges by weight: (A-B: 2), (A-D: 3), (C-D: 4), (B-D: 5), (A-C: 6).
• Start by adding the smallest edge (A-B: 2).
• Add the next smallest edge (A-D: 3).
• Add (C-D: 4).
• The edges (B-D: 5) and (A-C: 6) would form cycles, so they are excluded.
• The MST is formed by edges: A-B (2), A-D (3), C-D (4). The total weight is 2+3+4=92 + 3 + 4 = 9.

2. Prim’s Algorithm:

Prim’s algorithm is another greedy algorithm, but it works by growing the MST one edge at a time, starting from an arbitrary vertex. It selects the minimum weight edge that connects a vertex inside the MST to a vertex outside the MST.

Steps of Prim’s Algorithm:

1. Start with an arbitrary vertex as the MST.
2. Find the smallest edge that connects a vertex inside the MST to a vertex outside the MST.
3. Add that edge and vertex to the MST.
4. Repeat the process until all vertices are included in the MST.

Example (using the graph above):

• Start at vertex A.
• The smallest edge from A is (A-B: 2), so we add that edge.
• From A and B, the smallest edge is (A-D: 3), so we add that edge.
• From A, B, and D, the smallest edge is (C-D: 4), so we add that edge.
• The MST is formed by edges: A-B (2), A-D (3), C-D (4). The total weight is 2+3+4=92 + 3 + 4 = 9.

3. Boruvka’s Algorithm:

Boruvka’s algorithm is a less common but effective algorithm, especially for distributed computing. It works by finding the cheapest edge for each connected component and merging components progressively.

Applications of Minimum Spanning Tree:

1. Network Design:
   • Telecommunication Networks: Minimizing the cost of laying cables or connecting nodes in a communication network (e.g., connecting cities with the minimum cost).
   • Computer Networks: Minimizing the cost of connecting devices (routers, switches) in a network.
2. Cluster Analysis:
   • In data mining and machine learning, MSTs are used in hierarchical clustering to group similar items by minimizing the distance between them.
3. Approximating Solutions to Other Problems:
   • MSTs can be used to approximate solutions for problems like the Traveling Salesman Problem (TSP), where finding an optimal path is difficult, but the MST can serve as a good approximation.
4. Electrical Circuits:
   • Used in designing circuits where you need to minimize the total length of wires connecting components.

Complexity of MST Algorithms:

• Kruskal’s Algorithm: The time complexity is O(Elog⁡E)O(E \log E), where EE is the number of edges. The sorting step takes O(Elog⁡E)O(E \log E), and the union-find operations take O(α(V))O(\alpha(V)), where α\alpha is the inverse Ackermann function, which grows very slowly.
• Prim’s Algorithm: The time complexity is O(Elog⁡V)O(E \log V) using a binary heap or priority queue, where VV is the number of vertices, and EE is the number of edges.

Conclusion:

A Minimum Spanning Tree (MST) is an essential concept in graph theory that helps in minimizing the total weight of connecting all vertices in a graph. It’s useful in a variety of fields, especially network design and optimization problems. The two main algorithms to find an MST are Kruskal’s algorithm and Prim’s algorithm, both of which are based on greedy principles and can be applied to solve problems efficiently.