Now that we've seen how the Disjoint Set Union makes Kruskal's algorithm efficient, let's compare Prim's and Kruskal's side-by-side to understand their core differences and when to use each one.
- Prim's Algorithm is like a single-celled organism growing. It starts with one vertex and greedily expands its single tree by adding the cheapest edge connecting a vertex in the tree to a vertex outside the tree.
- Kruskal's Algorithm is like a nation builder. It starts with a "forest" of individual vertices and repeatedly connects the two closest, distinct components with the cheapest possible edge, until only one component (the MST) remains.
- The choice often depends on graph density. For a dense graph where $E \approx V^2$, Prim's using an adjacency matrix can achieve $O(V^2)$, which is faster. For sparse graphs, their complexities are similar ($O(E \log V)$), but Kruskal's can be simpler to implement.
| Feature | Prim's Algorithm | Kruskal's Algorithm |
|---|---|---|
| Fundamental Approach | Grows one connected tree from a starting vertex. | Connects a forest of components until one remains. |
| Key Data Structure | Priority Queue (to find the cheapest frontier edge). | Disjoint Set Union (to detect cycles). |
| Complexity | $O(E \log V)$ (using a binary heap). | $O(E \log E)$ or $O(E \log V)$ (dominated by edge sorting). |
| Graph Density | Excellent for dense graphs, especially the $O(V^2)$ array-based version. | Often preferred for sparse graphs. |
| Process | Picks the best edge connected to the current tree. | Picks the best edge in the entire graph that doesn't form a cycle. |