Is there any restriction on selecting path for the MaxFlow problem in datastructure?

Question

In the following max flow problem, the algorithm could choose S-A-D-T path first. In that case, the algorithm will not see any more augmenting paths so it will produce 4 as the answer to the maximum flow. However, if the algorithm would select any other path first, then it would see that the max flow becomes five.

Answer 1

It seems you did not notice that Ford-Fulkerson algorithm updates the residual graph in two ways:

1. subtract path flow from all edges along the selected path
2. add path flow along the reverse edges of the selected path

Therefore after you have selected the S-A-D-T path with flow 3, the residual graph will now contain an augmenting path S-B-D-A-C-T with flow 2, thus reaching a total flow of 5. Flow along the reverse edges just cancels the existing flow and thus only flow of 2 will remain from A to D.