Suppose P is the shortest path from a starting node s to a destination node v in a graph G. Suppose also that G' is G with all edge weights modified by multiplying a constant factor C > 0.Then P (with the corresponding edge weight modifications) is also the shortest path between s and v in the modified graph G'.

Yes, the statement is correct. The shortest path from node s to node v in graph G will remain the same in the modified graph G' even after all edge weights are multiplied by a constant factor C > 0. This is because multiplying all edge weights by a constant factor does not change the relative weights of the edges. Therefore, the shortest path in G will still have the smallest total weight in G', preserving the property of the shortest path.

Here are the steps to understand this:

1. Let's say the shortest path P in graph G from s to v is through nodes s, a, b, ..., v with weights w1, w2, ..., wn respectively. So, the total weight of path P is w1 + w2 + ... + wn.

2. Now, in the modified graph G', the weights of the edges in path P become Cw1, Cw2, ..., Cwn. So, the total weight of path P in G' is Cw1 + Cw2 + ... + Cwn = C * (w1 + w2 + ... + wn).

3. As you can see, the total weight of path P in G' is just the total weight of path P in G multiplied by the constant factor C.

4. Since all edge weights in G' are multiplied by the same constant factor C, the relative weights of the edges remain the same. Therefore, if path P was the shortest path in G, it will still be the shortest path in G' because its total weight is still the smallest.

5. Hence, the statement is correct.