Graphs 101: Minimum Spanning Tree (MST)

In the previous parts of our Graph Series, we studied the shortest path algorithm, which is **Floyd–Warshall**, which helps find minimum distances between nodes in a graph.

In this part, we move to Minimum Spanning Trees, focusing on how to connect all vertices with minimum total cost, laying the foundation for algorithms like Prim’s Algorithm.

So let’s get started….

Why Minimum Spanning Tree Matters

In many real-world systems, the goal is not shortest paths but minimum cost connectivity.

For example:

• Connecting cities with minimum total road length
• Designing computer networks with minimum cable cost
• Electrical wiring systems with minimum material usage

In all these cases:

• Every node must be connected
• Cycles are unnecessary
• Cost must be minimized

This is exactly the problem solved by Minimum Spanning Trees.

What Problem Does Minimum Spanning Tree Solve?

The Minimum Spanning Tree helps us:

• Connect all the vertices in a graph
• Avoid cycles
• Minimize the total edge weight
• Ensure the graph remains connected using the fewest edges

Unlike shortest path algorithms, MST:

• Does not focus on the distance between two specific nodes
• Focuses on the overall structure and cost

Type of Graph Used

Minimum Spanning Tree works on:

• Connected graphs
• Weighted graphs
• Typically undirected graphs

If the graph is disconnected, a spanning tree cannot be formed.

What Is a Spanning Tree?

Before understanding MST, we must understand a Spanning Tree.

A Spanning Tree is a subgraph that:

• Includes all vertices of the original graph
• Is connected
• Contains no cycles
• Has exactly V − 1 edges, where V is the number of vertices

A graph can have multiple different spanning trees.

What Is a Minimum Spanning Tree?

A Minimum Spanning Tree is defined as:

A spanning tree whose total edge weight is minimum among all possible spanning trees of the graph.

In simple terms:

• We try all valid spanning trees
• We calculate the sum of their edge weights
• The one with the lowest total cost is the MST

Can There Be More Than One MST?

Yes.

If multiple spanning trees have the same minimum total weight, the graph can have more than one Minimum Spanning Tree.

Key Difference from Shortest Path Algorithms

Shortest Path Algorithms

• Focus on finding the minimum distance between nodes
• Usually start from a single source node
• Optimize distance or path length
• Examples: Dijkstra’s Algorithm and Bellman–Ford Algorithm

Minimum Spanning Tree Algorithms

• Focus on connecting all vertices in the graph
• Do not use a source node
• Optimize the total cost of connections
• Examples: Prim’s Algorithm and Kruskal’s Algorithm

Why We Need Special Algorithms for MST

Although many spanning trees are possible, manually checking all of them is inefficient.

To solve this efficiently, we use:

• Kruskal’s Algorithm
• Prim’s Algorithm

Both algorithms are greedy and widely used in interviews and real-world systems.

Key Takeaways

• A spanning tree connects all vertices without cycles using V − 1 edges
• A Minimum Spanning Tree minimizes the total edge weight
• MST is used in cost-optimization problems
• MST is different from shortest path algorithms
• Efficient MST algorithms are essential for large graphs

What’s Next?

So far in our Graph Series, we have covered single-source shortest paths, all-pairs shortest paths, and the basics of cost-efficient graph connectivity.

In the next part, we will focus on Prim’s Algorithm, understanding how it builds a Minimum Spanning Tree, its intuition, implementation, and when to use it over Kruskal’s Algorithm.Stay tuned 😄