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minimum possible total edge weight. In simpler terms, it is the cheapest way to connect all

points in a network. MSTs are widely used in designing efficient communication networks,

electrical grids, transportation systems, and clustering algorithms in machine learning.
The concept of MST is important because it ensures connectivity with minimal cost.

For example, if we want to connect several cities with roads, the MST provides the blueprint

for building roads that connect all cities while minimizing construction costs.

Two of the most famous algorithms for finding MSTs are Prim’s Algorithm and

Kruskal’s Algorithm. Both guarantee the minimum spanning tree but approach the problem
differently.

8.5.1 Prim’s Algorithm

Prim’s algorithm builds the MST by starting from a single vertex and repeatedly adding

the smallest edge that connects a new vertex to the growing tree. It uses a greedy approach,
always choosing the minimum weight edge available at each step.

Step-by-step process:

1. Start with any vertex as the initial tree.
2. At each step, add the smallest edge that connects a vertex in the tree to a vertex outside

the tree.

3. Repeat until all vertices are included.

Prim’s algorithm is efficient for dense graphs and can be implemented using a priority

queue to select the smallest edge quickly.
Example in C++:

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