1 / 48

Flow Networks

Flow Networks. zichun@comp.nus.edu.sg. Formalization. Basic Results. Min-cut. Ford-Fulkerson. Edmunds-Karp. Bipartite Matching. Flow Network. Directed Graph G = (V, E) Each edge has a capacity. Properties of Flow. Capacity Constraint Skew Symmetry Flow Conservation. Maximum Flow.

thuyet
Télécharger la présentation

Flow Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Flow Networks zichun@comp.nus.edu.sg

  2. Formalization Basic Results Min-cut Ford-Fulkerson Edmunds-Karp Bipartite Matching

  3. Flow Network • Directed Graph G = (V, E) • Each edge has a capacity

  4. Properties of Flow Capacity Constraint Skew Symmetry Flow Conservation

  5. Maximum Flow • Returns Maximum Flow of G Value of Flow

  6. Motivating Problem 12 v1 v3 20 16 Factory Warehouse s t 10 4 7 9 13 4 v2 v4 14

  7. Motivating Problem 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14

  8. 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14

  9. Multiple Sources / Sinks 12 v1 v3 20 16 t1 s1 10 4 7 9 13 4 4 s2 t2 v2 v4 4 14

  10. Multiple Sources / Sinks 12 v1 v3 20 16 t1 s1 10 4 7 9 13 t S 4 4 s2 t2 v2 v4 4 14

  11. Implicit Summation Notation

  12. Key Equalities Let G = (V, E) be a flow network, and let f be a flow in G. Then

  13. Capacity Constraint Skew Symmetry Flow Conservation Homomorphism

  14. Flow Value Prove Definition Homomorphism Flow Conservation Skew Symmetry Homomorphism Flow Conservation

  15. Ford-Fulkerson flow=0while(augmenting path p)) { augment flow f along p } return flow

  16. Residual network Induced network from G=(V, E) and flow f

  17. 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14 v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3

  18. Augmenting Path A pathof non-zeroweight from s to t in Gf

  19. 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14 v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3

  20. 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14 v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3

  21. 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14 v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3

  22. 12/12 v1 v3 11/16 20/20 Factory Warehouse s t 10 1/4 7/7 0/9 13/13 4/4 v2 v4 11/14 v1 0 v3 0 11 12 Factory Warehouse 5 20 9 s t 3 11 7 0 0 13 0 0 11 v2 v4 4 3

  23. S-T Cut • A cut C=(S, T) of a flow network G=(V, E) is a partition of V into S and T = V – S • Net flow across • Capacity of cut:

  24. 12/12 v1 v3 11/16 15/20 Factory Warehouse s t 10 1/4 7/7 4/9 8/13 4/4 v2 v4 11/14 f(S, T) = 12 – 4 + 11 = 19c(S, T) = 12 + 14 = 26

  25. Let f be a flow in a flow network G with source s and sink t, and let (S, T) be a cut of G. Then the net flow across (S, T) is f(S, T) = |f| Homomorphism Flow Conservation Homomorphism Flow Conservation Definition

  26. The value of any flow f in a flow network G is bounded by the capacity of any cut of G

  27. Min-Cut Max-Flow • f is a maximum flow in G • The residual network Gf contains no augmenting path • |f|= c(S, T) for some cut (S, T) of G

  28. Premise: f is a max-flow in G • Assume Gf has augmenting path p • We can augment Gf with p to get a flow f’ > f • Contradicts [1] • Hence Gf has no augmenting paths

  29. Premise: Gf has no augmenting paths • Let • T = V-S • (S, T) is a cut • Otherwise

  30. The value of any flow f in a flow network G is bounded by the capacity of any cut of G • , |f| is maximum

  31. Ford-Fulkerson flow=0while(augmenting path p)) { augment flow f along p } return flow Termination: Gf has no augmenting path iff flow is maximum

  32. Run-time flow=0while(augmenting path p)) { augment flow f along p } return flow O(E|f*|) where |f*| is the maximum flow

  33. v1 1,000,000 1,000,000 s 1 t 1,000,000 1,000,000 v2

  34. v1 1,000,000 1,000,000 s 1 t 1,000,000 1,000,000 v2

  35. v1 999,999 1,000,000 1 s 1 t 1 1,000,000 1,000,000 v2

  36. v1 999,999 1,000,000 1 s 1 t 1 1,000,000 1,000,000 v2

  37. v1 999,999 999,999 1 1 s 1 t 1 1 999,999 1,000,000 v2

  38. v1 999,999 999,999 1 1 s 1 t 1 1 999,999 1,000,000 v2

  39. Edmunds Karp flow=0while(augmenting path p)) { find augmenting path by BFS augment flow f along p } return flow Run-time: O(VE2)

  40. v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3 Critical Edge

  41. Lemma: Edges can be critical at most times v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3 Critical Edge

  42. v1 0 v3 5 11 12 Factory Warehouse 5 15 5 s t 3 11 7 0 4 8 0 5 11 v2 v4 4 3

  43. Assume (u, v) is a critical edge • Augmenting-paths are shortest path [by EK] • [from 1] • After flow-augmentation, cf(u,v) = 0 • (u, v) cannot be critical path until another augmenting path containing (v, u). • Let the flow at this point be f’ • i.e, each time (u, v) is augmenting-path, path length increases by 2 • (u, v) can be critical edge at most times

  44. Lemma: Edges can be critical at most times Since there are O(E) edges, the number of augmenting path is bounded by O(VE) [by Lemma]. Run-time: O(VE2)

  45. Bipartite Matching

  46. Bipartite Matching

  47. Bipartite Matching MaxFlow s t

  48. Bipartite Matching MaxFlow s t

More Related