I am trying to find out the time complexity of finding a residual network after sending $f$ units of flow from a source node to a sink node. According to Ford-Fulkerson algorithm, I guess that it should be $O(mf)$ where $m$ is the number of arcs (edges) in the network and $f$ is the amount of transmitted flow from source node to sink node. Here is some more explanation on my guess:
In fact, to find the residual network after sending $f$ units of flow from a source node to a sink node, we need exactly to transmit these $f$ units of flow (data) through the network and update the arcs' capacities to get the residual network. So, (1) I start with the initial network, (2) find an available path from the source to the sink node, (3) send one units of data through the path, (4) update the arcs' capacities, and then go back to stage (2) if the transmitted flow is still less than $f$.
Time complexity of the method:
The time complexity of finding a path in the network is $O(m)$, and since I am doing this $f$ times, the time complexity of the method is $O(mf)$.
question:
Now, my question is that if my method is the best method (according to time complexity) or there are some technique with the time complexity being less than $O(mf)$.
I highly appreciate any help in advance.