Understanding Spanning Trees and Minimum Cost Spanning Trees: Prim's and Kruskal's Algorithm Explained and Their Differences

A spanning tree is like a snapshot of a connected graph, capturing the essence of its undirected and interconnected nature.

Minimum Cost Spanning Tree

In the realm of graphs, a minimum-weight tree emerges. This tree, within a weighted graph, gracefully encompasses all vertices while minimizing the collective weight of its edges. In simpler terms, it's the tree where the sum of edge weights is kept at its minimum.

Unpacking Prim's Algorithm

Prim's Algorithm, a greedy strategy, becomes our guide in unraveling the minimum spanning tree from a graph. This algorithm meticulously selects edges to form a subset, ensuring each vertex is part of this orchestrated dance.

1. Initialization: Begin by choosing a vertex randomly, kickstarting the Minimum Spanning Tree (MST).

2. Connecting the Dots: Identify edges linking the tree to new vertices from the previous step. Out of these edges, cherry-pick the one with the minimum weight, seamlessly adding it to the growing tree.

3. Traversal Mastery: If the edge elegantly connects back to previous vertices, meticulously sift through to pinpoint the minimum edge. Once found, it gracefully joins the expanding tree.

4. Iterative Harmony: Rinse and repeat the second step until the symphony of edges constructs the desired Minimum Cost Spanning Tree.

In essence, Prim's Algorithm orchestrates an elegant dance of edges, selecting and incorporating them into the Minimum Spanning Tree until the entire graph is harmoniously encapsulated.