the first path we found was if we push as much flow as possiblethen we end up with the followingnotes figure we have pushed flow along the path / / / / / now we have run into problemour only options left are to push unit of water along the path and unit of water along after thatwe won' be able to find any more path from to tyetwe have only found flowwhich is not he maximum flow thusthe idea is not optimal since it depends on how we picked our path while we can try to find "goodpathpicking algorithmit would be nice if the algorithm is not dependent on the paths we chose crucial observation is that there is actually another path from to other than the two that we mentioned abovesuppose we redirect units of water from to tthis will decrease the amount of water running through the pipe (atto now we have path from in which we can flow units of waterthe graph now looks as followsfigure the bolded edge indicate which pipe has their flow adjusted / / / / / it is now easy to see that we can push unit of water along to obtain the maximum flow of if we carefully study in the figure what we have essentially done is to flow unit of water along aand then push back unit of water from band finally redirect the pushed back flow along to the sink so the key to completing the algorithm is the idea of pushing back flow if we have units of water flowing in the pipe (uv)then we can pretend there is pipe (vuwith capacity when we are trying to find path from to this is the concept of residual graph the residual graph of network flow is essentially the network graph except that for every edge (uvthat currently carries unit of waterthere is lovely professional university