Can you tell me some ideas for an efficient algorithm on this problem?

We have a directed graph and a function of weight w we know that there is only one edge with a negative weight and that there is no cycle in the graph with negative weight we have s a vertex
We have to find an algorithm that find the length of all the shortest paths from s to every vertex of G

Dijkstra's algorithm doesn't allow negative-weight edges, but you can still apply it to this problem. Let (i, j) be the negative-weight edge. Then:

1. remove edge (i, j) from the graph.
2. Run Dikstra's algorithm using s as the source.
3. Run Dikstra's algorithm using j as the source.
4. For each vertex v, there are two possibilities:
   1. The shortest path from s to v does not use edge (i,j):
      In this case, the results from step 2 give you the shortest path.
   2. The shortest path from s to v goes through edge (i, j)
      In this case, the shortest path is the shortest path from s to i (obtained in step 2) plus edge (i, j) plus the shortest path from j to v (obtained in step 3).

Thus you need to check both possibilities for every vertex and see which is better.

This is the fastest way I can think of off the top of my head. Bellman-Ford allows for negative-weight edges, but it's not as efficient as Dikstra.