Prim’s vs. Kruskal’s Algorithm: Understanding MST Algorithms

When working with graphs, finding the Minimum Spanning Tree (MST) is crucial for optimizing network designs, reducing costs, and improving efficiency. Two of the most popular algorithms for this are Prim’s Algorithm and Kruskal’s Algorithm. Let’s explore their differences, advantages, and use cases.

🔹 Prim’s Algorithm

✅ Approach: Starts with a single vertex and expands by adding the smallest edge that connects to the growing tree.
✅ Best for: Dense graphs (many edges).
✅ Time Complexity: O(E log V) (using priority queue).
✅ Greedy Strategy: Adds the nearest vertex at each step.

🔹 Kruskal’s Algorithm

✅ Approach: Sorts all edges by weight and adds the smallest edge (avoiding cycles) until all vertices are connected.
✅ Best for: Sparse graphs (fewer edges).
✅ Time Complexity: O(E log E) (due to sorting).
✅ Greedy Strategy: Always picks the smallest edge first.

📌 Key Differences

Feature	Prim’s Algorithm	Kruskal’s Algorithm
Approach	Vertex-based	Edge-based
Graph Type	Works better for dense graphs	Works better for sparse graphs
Sorting Required?	No	Yes (edges sorted by weight)
Data Structure	Priority Queue (Heap)	Disjoint Set (Union-Find)

🛠️ Use Cases

🔸 Prim’s Algorithm: Used in network routing, designing electrical circuits, and clustering in AI.
🔸 Kruskal’s Algorithm: Used in road networks, railway designs, and image segmentation.