1. Honors Track: Competitive Programming & Problem Solving Push-Relabel Algorithm Claire Claassen

2. Graph algorithms Structure  Max-flow problem  Recap augmenting path algorithms  Push relabel algorithm  Pseudocode  Example  Prove  Run-time

3. Max-flow problem Maximum flow problem Intuition: Given a network of pipes and a source and sink, how much water can be transported from source to sink?

4. Max-flow problem Bipartite graphs: Model relations between two classes of objects Examples Boys and girls: “boy x likes girl y” A B

5. Max-flow problem Bipartite graphs: Model relations between two classes of objects Examples Boys and girls: “boy x likes girl y” Players and clubs: “player x wants to play in club y” A B

6. Max-flow problem Bipartite graphs: Model relations between two classes of objects Examples Boys and girls: “boy x likes girl y” Players and clubs: “player x wants to play in club y” Employees and jobs: “employee x can perform job y” A B

7. Max-flow problem Maximum flow problem Input: A graph with edge capacities c ij and a source s and sink t Output: The maximum flow f ij satisfying the flow constraints Flow constraints 0 ≤ f ij ≤ c ij “flow is within capacity” Σ k f ki = Σ k f ik for all i ≠ s, t “flow in = flow out” s = 0 t = 5 1 2 3 4 c 01 c 02 c 12 c 13 c 14 c 24 c 35 c 45 c 43

8. Augmenting path algorithm Idea: Find path from s to t and increase flow on the entire path s = 0 t = 5 1 2 3 4 0/3 0/2 0/1 0/2 0/3 0/2 0/1

9. Augmenting path algorithm Idea: Find path from s to t and increase flow on the entire path s = 0 t = 5 1 2 3 4 0/3 0/2 0/1 0/2 0/3 0/2 0/1

10. Augmenting path algorithm Idea: Find path from s to t and increase flow on the entire path s = 0 t = 5 1 2 3 4 2/3 0/3 2/2 0/2 0/1 2/2 0/3 2/2 0/1

11. Augmenting path algorithm Idea: Find path from s to t and increase flow on the entire path s = 0 t = 5 1 2 3 4 2/3 0/2 2/2 0/1 2/2 2/3 2/2 0/1

12. Augmenting path algorithm Idea: Find path from s to t and increase flow on the entire path s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 3/3 2/2 1/1 Finish: no augmenting path left

13. Max-Flow idea General idea:  The flow is optimal if there’s no path between s and t Graph approach:  DFS ➨ Ford-Fulkerson algorithm with O(E f*) running time  BFS ➨ Edmonds-Karp algorithm with O(V E 2 ) running time Without searching for path  Push-relabel algorithm ➨ O(V 2 E)

14. Push-relabel approach

15. Pseudocode Pseudo-code:  h[s] = |V|  For each v V – {s} do h[v] = 0  For each (s,v) E do f(s,v) = c(s,v)  While f is not feasible flow If there’s a vertex v V – {s,t} and a vertex w V with e(v) > 0and h[v] > h[w] and c’(v,w) > 0  Push min(c’(v,w), e(v)) over edge (v,w)  Else h[v] = h[v] +1

16. Example HeightExcess flow 11 10 9 8 7 6s 5 4 3 2 1 01 2 3 4 t1 2 3 4 s = 0 t = 5 1 2 3 4 0/3 0/2 0/1 0/2 0/3 0/2 0/1

17. Example s = 0 t = 5 1 2 3 4 3/3 0/2 0/1 0/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 31 2 2 1 01 2 3 4 t3 4

18. Example s = 0 t = 5 1 2 3 4 3/3 0/2 0/1 0/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 31 2 2 11 02 3 4 t3 4

19. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 0/1 0/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 32 23 111 02 3 4 t4

20. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 0/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 32 23 114 02 3 4 t1

21. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 0/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 32 23 11 24 03 4 t1

22. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 34 23 11 22 03 4 t1

23. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 34 223 112 03 4 t1

24. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 4 324 23 112 03 4 t1

25. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 5 42 34 23 112 03 4 t1

26. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 52 4 34 23 112 03 4 t1

27. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 7 6s 2 5 4 34 23 112 03 4 t1

28. Example s = 0 t = 5 1 2 3 4 3/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 34 23 112 03 4 t1

29. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 34 23 11 03 4 t1 2

30. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 0/3 0/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 34 23 11 3 04 t1 2

31. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 2/3 0/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 34 2 11 3 04 t1 2 3

32. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 2/3 0/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 34 2 11 3 4 0t1 2 3

33. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 2/3 2/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 3 2 11 3 44 0t1 2 3

34. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 2/3 2/2 0/1 HeightExcess flow 11 10 9 8 72 6s 5 4 3 24 11 34 0t1 2 3

35. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 2/3 2/2 1/1 HeightExcess flow 11 10 9 8 72 6s 5 4 3 24 11 33 0t1 2 4

36. Example s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 3/3 2/2 1/1 HeightExcess flow 11 10 9 8 72 6s 5 4 3 24 11 3 0t1 2 3 4

37. Comparison s = 0 t = 5 1 2 3 4 3/3 2/3 0/2 2/2 1/1 2/2 3/3 2/2 1/1 s = 0 t = 5 2 4 3/3 2/3 0/2 1/1 2/2 3/3 2/2 1/1

38. Prove NodeExcess flow 0-3 13 20 30 s = 0 1 t=3 2 3/3 0/2 0/1

39. Prove s = 0 t = 5 1 2 3 4 3/3 0/2 0/1 0/2 0/3 0/2 0/1

40. Prove 0 1 2

41. Run time

42. Run time 2

43. Run time comparison Ford-Fulkinson s = 0 1 2 0/1000 0/1 0/1000

44. Run time comparison Ford-Fulkinson s = 0 1 2 0/1000 0/1 0/1000

45. Run time comparison Ford-Fulkinson s = 0 1 2 1/1000 0/1000 1/1 0/1000 1/1000

46. Run time comparison Ford-Fulkinson s = 0 1 2 1/1000 0/1000 1/1 0/1000 1/1000

47. Run time comparison Ford-Fulkinson s = 0 1 2 1/1000 0/1 1/1000

48. Run time comparison Ford-Fulkinson s = 0 1 2 1/1000 0/1 1/1000

49. Run time comparison Ford-Fulkinson s = 0 1 2 2/1000 1/1000 1/1 1/1000 2/1000

50. Run time comparison Ford-Fulkinson s = 0 1 2 2/1000 1/1000 1/1 1/1000 2/1000

51. Run time comparison Ford-Fulkinson s = 0 1 2 2/1000 0/1 2/1000

52. Run time comparison Ford-Fulkinson s = 0 1 2 2/1000 0/1 2/1000

53. Run time comparison Ford-Fulkinson s = 0 1 2 3/1000 2/1000 1/1 2/1000 3/1000

54. Run time comparison Ford-Fulkinson s = 0 1 2 3/1000 2/1000 1/1 2/1000 3/1000

55. Run time comparison Ford-Fulkinson s = 0 1 2 3/1000 0/1 3/1000

56. Run time comparison Ford-Fulkerson Edmonds-Karp s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 0/1000 0/1 0/1000

57. Run time comparison Ford-Fulkerson Edmonds-Karp s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 0/1000 0/1 0/1000

58. Run time comparison Ford-Fulkerson Edmonds-Karp s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1000 0/1 1000/1000 0/1000

59. Run time comparison Ford-Fulkerson Edmonds-Karp s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1000 0/1 1000/1000 0/1000

60. Run time comparison Ford-Fulkerson Edmonds-Karp s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000

61. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 0/1000 0/1 0/1000

62. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 0/1000

63. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 0/1000

64. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 1/1 0/1000

65. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 1/1 999/1000 0/1000

66. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 1/1 999/1000 0/1000

67. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 1/1 999/1000 1000/1000

68. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 999/1000 1000/1000

69. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 999/1000 1000/1000

70. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000

71. Run time comparison Ford-Fulkerson Edmonds-Karp Push-Relabel Algorithm s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000

72. Run time comparison s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000 s = 0 1 2 1000/1000 0/1 1000/1000