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I am trying to find a network, where Dinic's algorithm makes $|V|^2*|E|$ steps. Clearly it cannot be a network with $3$ or less vertices, but I am not able to come up with a working example for quite a while now.

Any idea?

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  • $\begingroup$ What exactly is a "step" to you? Also, if you want to gain any form of helpful understanding, you'll have to come up with a pattern that scales to arbitrary size; finite example sets don't have any bearing on $O$-asymptotics. $\endgroup$ – Raphael Nov 13 '18 at 15:00
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  1. Dinic's algorithm runs in $O(|V|^2 \cdot |E|)$ time, so it might never make $|V|^2 \cdot |E|$ steps.

  2. Finding a single example is insufficient, as any finite-size counterexample can be "hidden" as a constant in asymptotic notation. You will need an infinite sequence of networks of strictly increasing size with a lower time bound of $\Omega(|V|^2 \cdot |E|)$ to prove that the bound is indeed tight.

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  • $\begingroup$ Obviously, a construction parameterized by |V| and |E| can suffice, instead of a single example. $\endgroup$ – Gassa Jun 16 '18 at 8:59

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