15.082 and 6.855J The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow Problem Obtain a network, and use the same network to illustrate the.

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15.082 and 6.855J The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow Problem Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

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.

Initialize Distances 2 2 1 5 2 s 1 4 t 1 3 4 5 4 3 2 1 1 3 1 1 2 4 3 2 5 4 3 2 1 1 3 1 1 2 4 2 s 1 4 t 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)

Saturate Arcs out of node s 3 s 4 2 2 2 5 1 5 4 3 2 1 1 3 1 1 2 4 6 2 s 1 4 1 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.

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

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

Select, then relabel/push 3 s 4 2 2 2 1 5 5 4 3 2 1 1 3 1 1 2 4 2 s 6 1 1 4 t 3 2 s 3 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

Select, then relabel/push 3 3 s 1 4 2 2 2 2 2 1 1 5 5 4 3 2 1 3 1 3 1 1 2 4 6 2 s 4 1 1 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

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

Select, then relabel/push 2 3 s 4 1 2 2 2 2 5 2 1 1 5 4 3 2 1 3 1 3 1 1 2 4 6 2 s 4 1 1 t 3 2 3 s s 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.

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

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

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

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

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

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

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

Select, then relabel/push 1 s 1 4 4 2 4 2 2 2 5 3 3 5 2 1 2 3 5 4 3 2 1 5 3 1 3 1 2 1 2 4 5 5 6 2 s 1 4 1 t 3 3 s s 4 3 4 2 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.