Here you can find the following lemma with its proof.Lemma: In a network with unit edge capacities,Dinitz algorithm terminates after $O(\sqrt{m})$blocking flow computations.Proof:Consider the layered network after $\sqrt{m}$ iterations.It has at most m edges. The average number ofedges between any two layers is $\frac{m}{\sqrt{m}} = \sqrt{m}$.Hence, there must be at least one layer $i$ with at most $\sqrt{m}$ edges. Now consider the cut$S=\{v | level_f(v) < i\}$. It has capacity at most $\sqrt{m}$.Since each blocking flow augmentation increases flowby at least one, after $\sqrt{m}$ additional blockingflow augmentations,  a maximum flow has been found.Hence, no more than 2$\sqrt{m}$ = $O(\sqrt{m})$blocking flow augmentations are needed.I don't understand the proof. The layered graph $G_{L}$ has m edges. But how does it look like after $\sqrt{m}$ iterations? Is the average number of edges of $G_{L}$ between any two layers $\frac{m}{\sqrt{m}} = \sqrt{m}$ after $\sqrt{m}$ iterations? And the confusion continuous with the set $S$.Could someone unfold this proof for better understanding? 
Do not understand a lemma regarding dinics algorithm
algorithms;graphs;optimization;network flow