Assuming value of every weight to be greater than 10, in which of the following cases the shortest path of a directed weighted graph from 2 vertices u and v will never change? ans. in both the cases of multiplying and adding by 10 add all values by 10 multiply all values by 10 subtract 10 from all the values

The shortest path of a directed weighted graph from 2 vertices u and v will never change in the cases of adding all values by 10 and multiplying all values by 10.

Here's why:

1. Adding all values by 10: When you add a constant value to all weights, the relative differences between the weights remain the same. Therefore, the shortest path doesn't change because it's determined by these relative differences, not the absolute values of the weights.

2. Multiplying all values by 10: Similarly, when you multiply all weights by a constant, the relative differences between the weights remain the same. Therefore, the shortest path doesn't change.

However, subtracting 10 from all the values could potentially change the shortest path. This is because subtracting a constant from all weights can change the relative differences between the weights, especially if some weights become negative as a result. In a graph with negative weights, the shortest path could potentially include a cycle, which wouldn't be the case if all weights were positive.