What is the best time complexity we can achieve to precompute all-pairs shortest paths in a weighted graph?

The best time complexity we can achieve to precompute all-pairs shortest paths in a weighted graph is O(V^3), where V is the number of vertices in the graph. This can be achieved using Floyd-Warshall algorithm.

Here are the steps:

1. Initialize a 2D array, dist[][], such that dist[i][j] holds the shortest distance from vertex i to vertex j. Initially, for all i and j, dist[i][j] is equal to the weight of the edge between i and j, or infinity if there is no direct edge between i and j.

2. For each vertex v, consider it as an intermediate vertex for all pairs of vertices (i, j). Update dist[i][j] to the minimum of its current value and the sum of the shortest paths from i to v and from v to j. This can be done with the following code:

   for v = 1 to V for i = 1 to V for j = 1 to V dist[i][j] = min(dist[i][j], dist[i][v] + dist[v][j])

3. After the above step, dist[i][j] will hold the shortest distance from i to j, considering all possible intermediate vertices.

This algorithm works because it gradually improves an estimate on the shortest path between two vertices, until it eventually reaches the actual shortest path. The time complexity is O(V^3) because there are three nested loops, each of which can iterate up to V times.