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

Preflow Push This is the original network, plus reversals of the arcs s t

s t Preflow Push This is the original network, which is also the original residual network s t

s t Initialize Distances s t The node label henceforth will be the distance label t 453 s2 d(j) is at most the distance of j to t in G(x)

s t Saturate Arcs out of node s s t Saturate arcs out of node s t 453 s2 Move excess to the adjacent arcs s s Relabel node s after all incident arcs have been saturated. 2 34

s t Select, then relabel/push s t Select an active node, that is, one with excess t 453 s2 Push/Relabel s s Update excess after a push 1 1

s t Select, then relabel/push s t Select an active node, that is, one with excess t 453 s s s No arc incident to the selected node is admissible. So relabel

s t Select, then relabel/push s t Select an active node, that is, one with excess t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s There is no incident admissible arc. So Relabel

s t Select, then relabel/push s t Select an active node t 4 5s s s Push/Relabel

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

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t Select an active node t 45 s s s Push/Relabel

s t Select, then relabel/push s t One can keep pushing flow between nodes 2 and 5 until eventually all flow returns to node s t 45 s s s There are ways to speed up the last iterations

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