Now that we've seen the efficient $O(E \log V)$ implementation, the natural question is: when is it actually better than the simpler $O(V^2)$ approach? The answer lies in the density of the graph.
- We can categorize graphs based on the relationship between their vertices $V$ and edges $E$:
- Sparse Graphs: The number of edges $E$ is much smaller than the maximum possible. For a sparse graph, $E$ is roughly proportional to $V$, or $E \approx O(V)$.
- Dense Graphs: The number of edges $E$ is close to the maximum possible. In a dense graph, $E$ is proportional to $V^2$, or $E \approx O(V^2)$.
- The choice of algorithm depends entirely on this distinction:
- For sparse graphs, $O(E \log V)$ simplifies to $O(V \log V)$, which is significantly faster than $O(V^2)$.
- For dense graphs, $O(E \log V)$ becomes $O(V^2 \log V)$, which is slower than the $O(V^2)$ implementation.
- Since most real-world problems—like computer networks, social networks, and transportation systems—are represented by sparse graphs, the $O(E \log V)$ algorithm is the industry standard for finding Minimum Spanning Trees.
| Graph Type | Edge Relationship | Adjacency Matrix Complexity | Adjacency List + Heap Complexity | Winner |
|---|---|---|---|---|
| Sparse Graph | E ≈ O(V) |
O(V²) |
O(V log V) |
Adj. List + Heap |
| Dense Graph | E ≈ O(V²) |
O(V²) |
O(V² log V) |
Adj. Matrix |