Floyd-Warshall algorithm utilizes __________ to solve the all-pairs shortest paths problem on a directed graph in __________ time.

The Floyd-Warshall algorithm utilizes dynamic programming to solve the all-pairs shortest paths problem on a directed graph in O(n^3) time.

Here are the steps:

1. Initialization: Create a matrix, call it dist[][], such that dist[i][j] is initialized as the weight of the edge between i and j for every edge in the graph. If there is no edge between i and j, then dist[i][j] is initialized as infinity. Also, initialize dist[i][i] as 0 for every vertex i.

2. Iteration: The algorithm then performs three nested loops, each one iterating through the vertices of the graph. The outermost loop goes through each vertex k, the middle loop goes through each vertex i, and the innermost loop goes through each vertex j.

3. Relaxation: In each iteration of the innermost loop, the algorithm checks if the shortest path from vertex i to vertex j can be improved by going through vertex k, i.e., if dist[i][j] > dist[i][k] + dist[k][j]. If this is the case, then it updates dist[i][j] to dist[i][k] + dist[k][j].

4. Result: After all iterations, dist[i][j] for all pairs of vertices i and j will be the shortest possible distance between them in the graph.

This algorithm runs in O(n^3) time because it performs a constant amount of work for each pair of vertices for each vertex in the graph, and there are n^3 such combinations if n is the number of vertices in the graph.