Dinic speed

The computational complexity of Dinic is O(V^2E), and when E is proportional to V, V=10000, it seems that it cannot be solved, but it is considered "faster in reality", and in fact it is fast, then how fast is it?

• Dinic method and its time complexity - Misawamemo

  • Computational complexity is reduced under some conditions when the side capacity is an integer.
    • When maximum flow is F $O(FE)$.
    • When the capacity of an edge is at most C $O(C E^{3/2})$.
      • and no multiple edges $O(CV^{2/3}E)$.
    • When the flow through each vertex is at most F $O(FV^{1/2}E)$.
      • Solving bipartite matching with maximum flow corresponds to the case F=1 above
  • For implementations using dynamic trees, for general graphs $O(VE\log V)$.
  • On the maximum flow problem. - Practice Book. - TopCoder Department
  • On the Maximum Flow Problem, Part 3 - Practice Chos. - TopCoder Department

• If the edge capacity is constant $O(\min { E^{1/2}, V^{2/3} } E)$ PDF

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