The Floyd-Warshall algorithm is a simple and important way to find the shortest paths between every pair of points in a weighted graph, especially when the graph has many connections. Here’s how to use this algorithm in easy steps.

Step 1: Create the Distance Matrix

First, we need a distance matrix, which we'll call $D$. This matrix shows the shortest distance from one point (vertex) to another:

• If there is a direct connection between point $i$ and point $j$, then set $D[i][j]$ to the weight (or cost) of that connection.
• If there is no connection, set $D[i][j]$ to infinity (which means there's no path).
• For every point $i$, set $D[i][i] = 0$ because the distance from a point to itself is zero.

This setup is important for calculating the shortest paths later.

Step 2: Update the Paths

Next, we will figure out the shortest paths using a series of updates. The core of this algorithm uses three loops to check if the path from $i$ to $j$ can be made shorter by going through another point $k$:

• For each point $k$ (considered as an intermediary),
  • For each point $i$,
    • For each point $j$:
      • Update the distance matrix with this rule: $D[i][j] = \min(D[i][j], D[i][k] + D[k][j])$ This means we check if it’s shorter to go directly from $i$ to $j$ or to go from $i$ to $k$ and then from $k$ to $j$. If the second option is shorter, we update the distance.

This method ensures every possible path is considered gradually reducing the distances.

Step 3: Dealing with Infinite Distances

While implementing the algorithm, it’s important to handle infinite distances correctly. If at any time $D[i][k]$ or $D[k][j]$ is infinity, it means there's no valid path between those points. So, if we encounter this situation while updating, we won't change the distance for that pair, keeping that infinity value.

Step 4: Show the Final Distance Matrix

After updating, the distance matrix $D$ will show the shortest paths between every pair of points. We should display this result nicely:

• Print the distance matrix clearly so everyone can see the shortest path distances between points:

$$D = \begin{bmatrix} D[0][0] & D[0][1] & ... & D[0][n-1] \\ D[1][0] & D[1][1] & ... & D[1][n-1] \\ \vdots & \vdots & \ddots & \vdots \\ D[n-1][0] & D[n-1][1] & ... & D[n-1][n-1] \end{bmatrix}$$

Step 5: Time Complexity

It's also good to know how long this algorithm takes to run, especially with bigger graphs. The Floyd-Warshall algorithm takes $O(n^3)$ time, where $n$ is the number of points. Each of the three loops in the algorithm runs in $O(n)$ time. This method can take longer with sparse graphs compared to other methods like Dijkstra's or Bellman-Ford, but it's great when we need to find paths between all pairs of points.

Step 6: Space Complexity

The space complexity of this algorithm is $O(n^2)$ because of the distance matrix. This can be an issue if memory is limited, especially with very large graphs. Make sure you have enough memory available when you work with this matrix.

Step 7: Final Thoughts and Tips

The Floyd-Warshall algorithm is well-liked, but think about whether it's the best choice for your specific needs. If your graph has negative weights, watch out for negative cycles, as they can make the path calculations confusing, and paths might not have proper definitions.

To make the algorithm work better, you can try different methods or combine this algorithm with others based on the type of graph you have. This could include using other shortest path algorithms if your graph isn’t dense or using parallel processing techniques if possible.

By following these steps, you can successfully use the Floyd-Warshall algorithm to find the shortest paths between all pairs of points in a weighted graph. Proper setup, updates, and attention to infinite distances will help you execute the algorithm accurately and efficiently. Understanding how the algorithm works is key to using it well in real-life situations.