We've identified that the simple array-based implementation has an $O(V^2)$ complexity due to its repeated linear scan. While this seems inefficient, this approach is surprisingly effective for dense graphs.
- A graph is considered dense when the number of edges, $E$, is close to the maximum possible number of edges. In this scenario, $E$ is on the order of $V^2$ (we write this as $E = O(V^2)$).
- Comparison: More advanced implementations of Prim's algorithm achieve a complexity of $O(E \log V)$.
- Dense Case: If we substitute $E$ with $V^2$, the advanced complexity becomes $O(V^2 \log V)$.
- Conclusion: In a dense graph, $O(V^2)$ is asymptotically faster than $O(V^2 \log V)$. The simpler adjacency matrix approach wins!
- The $O(V^2)$ implementation is not only faster for dense graphs but is also often simpler to code and can have lower memory overhead than more complex data structures like binary heaps.
| Graph Type | Edge Relationship | Simple Impl. (Array) | Advanced Impl. (Heap) | Preferred Method |
|---|---|---|---|---|
| Dense | $E \approx V^2$ | $O(V^2)$ | $O(V^2 \log V)$ | Simple Impl. |
| Sparse | $E \approx V$ | $O(V^2)$ | $O(V \log V)$ | Advanced Impl. |