The Floyd-Warshall algorithm is a strong tool for finding the shortest paths in graphs. However, it doesn't work well for every situation, and there are some important limits to know.

Complexity Issues

• Time Complexity: The Floyd-Warshall algorithm takes a lot of time to run, especially with big graphs. Its speed is measured as $O(V^3)$, where $V$ is the number of points (or vertices) in the graph. This means that as the number of points increases, the amount of time it takes to solve the problem grows very quickly. In comparison, Dijkstra’s algorithm can run faster at $O((V + E) \log V)$ for graphs that are not too crowded, and the Bellman-Ford algorithm works in $O(V \cdot E)$.

• Space Complexity: This algorithm also needs a lot of space to keep track of distances between points. It needs $O(V^2)$ space, which can be too much when you're dealing with big networks. If the memory gets full, it can cause problems.

When to Use It

• Negative Weights: One good thing about Floyd-Warshall is that it can deal with negative edge weights, which are connections that have a "cost" of less than zero. But, it doesn’t work with graphs that have negative cycles, where you can keep going around a loop to get shorter and shorter paths. This can lead to wrong answers. Finding negative cycles can make things even more complicated.

• Dense vs. Sparse Graphs: The algorithm works best with dense graphs, where there are many edges connecting the points, close to $V^2$. But if a graph is sparse, meaning it has few edges, other algorithms like Dijkstra’s or Bellman-Ford are usually better choices. So, while it might seem like a universal solution, it doesn't always fit well in every case.

Ways to Overcome Limitations

• Graph Preprocessing: Before using Floyd-Warshall, you can simplify the graph. This could mean removing edges or points that don’t matter. By doing this, you can help make the algorithm run faster and use less memory.

• Hybrid Approaches: Sometimes, using a mix of algorithms works better. For example, you could use Floyd-Warshall to figure out some initial distances and then switch to Dijkstra’s for specific questions. This way, you get the best of both worlds.

Conclusion

In summary, the Floyd-Warshall algorithm is a useful tool for some problems related to finding the shortest paths. However, it can be slow and take up too much space in larger and less dense graphs. It also struggles with negative cycles, which adds to its problems. Understanding the type of graph you're dealing with and how to prepare it can help, but the challenges of using Floyd-Warshall are still significant when trying to solve the shortest path problems.