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15.082 and 6.855J The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow Problem.
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15.082 and 6.855J The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow Problem
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15.082 and 6.855J
The Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow
Problem
4
114
21
2
3
3
1
s
2
4
5
3
t
Preflow Push
This is the original network, which is also the original residual network.
4
11
4
21
2
3
3
1
s
2
4
5
3
t
4
114
21
2
3
3
1
s
2
4
5
3
t
Initialize Distances
4
11
4
21
2
3
3
1
s
2
4
5
3
t
The node label henceforth will be the distance label.
0
5
4
3
2
1
t
4 53
s2
d(j) is at most the distance of j to t in G(x)
02
2
1
1
1
4
114
21
2
3
3
1
s
2
4
5
3
t
Saturate Arcs out of node s
4
11
4
21
2
3
3
1
s
2
4
5
3
t
Saturate arcs out of node s.
0
5
4
3
2
1
t
4 53
s2
Move excess to the adjacent arcs
02
2
1
1
1
3
2
12
13
6s
s
Relabel node s after all incident arcs have been saturated.
2
3 4
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
3
t
Select an active node, that is, one with excess
0
5
4
3
2
1
t
4 53
s2
Push/Relabel
02
2
1
1
1
3
2
12
13
6s
s
Update excess after a push
11
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
3
t
Select an active node, that is, one with excess
0
5
4
3
2
1
t
4 53
s202
2
1
1
1
3
2
12
1
6s
s
No arc incident to the selected node is admissible. So relabel.
11
23
3
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node, that is, one with excess
0
5
4
3
2
1
t
4 5
s202
2
1
1
3
2
26
s
s
Push/Relabel
1
2
3
21
31
2
3
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s202
2
1
1
3
2
26
s
s
Push/Relabel
1
2
3
21
31
2
3 31
32
2
1
5
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s202
2
1
12
26
s
s
Push/Relabel
1
2
3
21
31
2
31
3
2
1
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s202
2
1
12
26
s
s
There is no incident admissible arc. So Relabel.
1
2
3
21
31
2
31
3
2
12
5
5
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4
5s202
2
1
12
26
s
s
Push/Relabel
1
2
3
21
3
2
31
3
2
22
1
41
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4
5s202
2
1
12
26
s
s
There is no incident admissible arc. So relabel.
1
2
3
21
3
2
31
3
2
22
1
41
3
5
5
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s202
2
1
12
26
s
s
Push/Relabel
1
2
3
21
3
2
31
3
2
22
1
41
3
5
5
3
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s202
2
1
12
26
s
s
Push/Relabel
1
2
3
21
3
22
222
41
3
5
5
3
1
141
4
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s202
2
1
12
26
s
s
Push/Relabel
1
2
3
21
3
22
222
41
3
5
5
3
1
4
24
2
2
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s02
2
1
12
26
s
s
Push/Relabel
1
2
3
21
3
22
222
41
3
5
5
3
1
4
4
2
4
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
Select an active node.
0
5
4
3
2
1
t
4 5
s02
2
1
12
26
s
s
Push/Relabel
1
2
3
21
3
21
322
41
3
5
5
3
1
4
4
2
4 35
5
5
4
114
21
2
3
3
1
s
2
4
5
3
t
Select, then relabel/push
4
11
4
21
2
3
3
1
s
2
4
5
t
One can keep pushing flow between nodes 2 and 5 until eventually all flow returns to node s.
0
5
4
3
2
1
t
4 5
s02
2
1
12
26
s
s
There are ways to speed up the last iterations.
1
2
3
21
3
21
322
41
3
5
5
3
1
4
4
2
4 35
5
5
MITOpenCourseWarehttp://ocw.mit.edu
15.082J / 6.855J / ESD.78J Network OptimizationFall 2010
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