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15.082 and 6.855J

15.082 and 6.855J. The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow Problem. Preflow Push. 4. 2. 5. 1. 3. 1. 1. 2. 4. s. 4. t. 3. 2. 1. 3. This is the original network, and the original residual network. 5. 4. 3. 2. 1. 0. Initialize Distances. 4. 2. 2.

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15.082 and 6.855J

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  1. 15.082 and 6.855J The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow Problem

  2. Preflow Push 4 2 5 1 3 1 1 2 4 s 4 t 3 2 1 3 This is the original network, and the original residual network.

  3. 5 4 3 2 1 0 Initialize Distances 4 2 2 5 1 1 3 1 1 2 4 2 s 1 4 t 0 3 2 2 s 1 3 4 5 1 3 t The node label henceforth will be the distance label. d(j) is at most the distance of j to t in G(x)

  4. 5 4 3 2 1 0 Saturate Arcs out of node s 3 s 4 2 2 2 1 5 1 3 1 1 2 4 6 s 2 4 1 1 0 t 3 2 2 2 s s 2 1 3 3 4 4 5 1 3 1 t 3 Saturate arcs out of node s. Move excess to the adjacent arcs Relabel node s after all incident arcs have been saturated.

  5. 5 4 3 2 1 0 Select, then relabel/push 3 s 4 2 2 2 1 5 1 3 1 1 2 4 s 2 6 1 1 4 t 0 3 2 2 s s 2 1 3 4 5 1 3 1 1 t 3 1 Select an active node, that is, one with excess Push/Relabel Update excess after a push

  6. 5 4 3 2 1 0 Select, then relabel/push 3 s 4 2 2 2 1 5 1 3 1 1 2 4 2 6 s 1 4 1 t 0 3 2 s 3 s 2 1 3 3 4 5 2 2 1 3 1 1 t 1 Select an active node, that is, one with excess No arc incident to the selected node is admissible. So relabel.

  7. 5 4 3 2 1 0 Select, then relabel/push 3 s 4 2 2 2 1 5 1 3 1 1 2 4 s 6 2 1 1 4 t 0 3 2 3 s 3 s 3 2 1 4 5 2 2 2 2 t 1 1 Select an active node, that is, one with excess Push/Relabel

  8. 3 3 5 4 3 2 1 0 Select, then relabel/push 3 s 4 1 2 2 2 2 2 1 5 1 3 1 3 1 1 2 4 6 2 s 1 4 1 0 t 3 2 2 3 s s 3 2 1 4 5 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  9. 5 4 3 2 1 0 Select, then relabel/push 2 3 s 4 1 2 2 2 2 1 1 5 3 1 3 1 1 2 4 s 6 2 4 1 1 t 0 3 2 s 3 s 3 2 1 4 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  10. 5 4 3 2 1 0 Select, then relabel/push 3 2 s 4 1 2 2 2 2 5 1 2 1 3 1 3 1 1 2 4 6 2 s 1 4 1 t 0 3 2 s s 3 5 2 3 1 4 5 5 2 2 2 t 1 1 Select an active node. There is no incident admissible arc. So Relabel.

  11. 5 4 3 2 1 0 Select, then relabel/push 1 3 2 s 4 1 2 2 2 2 5 2 2 1 3 1 3 1 1 2 4 s 6 2 1 4 1 0 t 3 2 3 s s 5 3 2 4 1 4 2 2 2 t 1 1 Select an active node. Push/Relabel

  12. 5 4 3 2 1 0 Select, then relabel/push 3 1 2 s 4 1 2 2 2 2 1 2 2 3 5 3 1 3 1 1 2 4 5 6 s 2 1 1 4 0 t 3 2 s s 3 5 5 3 4 2 1 4 2 2 2 t 1 1 Select an active node. There is no incident admissible arc. So relabel.

  13. 5 4 3 2 1 0 Select, then relabel/push 3 1 2 s 4 1 2 2 2 2 2 1 2 3 3 5 3 1 3 1 1 2 4 5 2 s 6 1 1 4 t 0 3 2 3 s s 2 4 3 1 4 5 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  14. 5 4 3 2 1 0 Select, then relabel/push 1 s 4 2 2 2 2 2 5 2 3 1 3 2 2 1 3 1 1 2 4 5 6 s 2 1 4 1 1 1 0 t 3 2 s 3 s 2 3 4 4 1 4 4 5 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  15. 5 4 3 2 1 0 Select, then relabel/push 1 s 4 2 2 2 2 2 4 2 5 3 3 2 2 1 2 1 3 1 2 1 2 4 5 s 6 2 4 1 1 0 t 3 2 2 s s 3 2 3 4 4 1 4 5 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  16. 5 4 3 2 1 0 Select, then relabel/push 1 s 4 2 2 4 2 2 4 2 2 1 2 3 3 5 2 1 3 1 2 1 2 4 5 s 2 6 1 4 1 t 0 3 3 s s 2 3 4 4 1 4 5 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  17. 5 4 3 2 1 0 Select, then relabel/push 1 s 4 1 4 2 2 2 4 2 5 1 3 5 2 3 3 2 5 3 1 3 1 2 1 2 4 5 5 s 6 2 1 4 1 0 t 3 3 s s 4 3 2 4 1 4 5 5 2 2 2 t 1 1 Select an active node. Push/Relabel

  18. 5 4 3 2 1 0 Select, then relabel/push 1 s 4 1 4 2 2 2 4 2 5 1 3 5 2 3 3 2 5 3 1 3 1 2 1 2 4 5 5 s 6 2 1 4 1 0 t 3 3 s s 4 3 2 4 1 4 5 5 2 2 2 t 1 1 One can keep pushing flow between nodes 2 and 5 until eventually all flow returns to node s. There are no paths from nodes 2 and 5 to t, and there are ways to speed up the last iterations.

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