Lecture 16
55
Lecture 16 Max Flows 3 Preflow-Push Algorithms
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Lecture 16. Max Flows 3 Preflow-Push Algorithms. Review. The Ford Fulkerson Maximum Flow Algorithm. Begin x := 0; create the residual network G(x); while there is some directed path from s to t in G(x) do begin let P be a path from s to t in G(x); ∆ := δ (P); - PowerPoint PPT Presentation
Transcript of Lecture 16
Lecture 16
Max Flows 3
Preflow-Push Algorithms
Review
The Ford Fulkerson Maximum Flow Algorithm
Begin x := 0;
create the residual network G(x);
while there is some directed path from s to t in G(x) do
begin
let P be a path from s to t in G(x);
∆:= δ(P);
send ∆ units of flow along P;
update the r's;
End
end {the flow x is now maximum}.
e(3)=1
Exact distance: d(i) equals the length of the shortest path from node i to node t in G(x)
The Goldberg-Tarjan Preflow Push Algorithm for the
Maximum Flow Problem
Preflow Push
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s
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3
t
This is the original network, and the original residual network.
Initialize Distances
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t
The node label henceforth will be the distance label.
0
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t
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s2
d(j) is at most the distance of j to t in G(x)
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1
1
Saturate Arcs out of node s
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t
Saturate arcs out of node s.
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s2
Move excess to the adjacent arcs
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6s
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Relabel node s after all incident arcs have been saturated.
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3 4
Select, then relabel/push
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t
Select an active node, that is, one with excess
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t
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s2
Push/Relabel
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s
Update excess after a push
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Select, then relabel/push
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t
Select an active node, that is, one with excess
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t
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s202
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1
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6s
s
No arc incident to the selected node is admissible. So relabel.
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3
Select, then relabel/push
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s
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t
Select an active node, that is, one with excess
0
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t
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s202
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s
s
Push/Relabel
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3
Select, then relabel/push
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s
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t
Select an active node.
0
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t
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s202
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1
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s
s
Push/Relabel
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5
Select, then relabel/push
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s
2
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5
t
Select an active node.
0
5
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t
4 5
s202
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1
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26
s
s
Push/Relabel
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2
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31
2
31
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2
1
Select, then relabel/push
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11
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21
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1
s
2
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5
t
Select an active node.
0
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t
4 5
s202
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1
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s
s
There is no incident admissible arc. So Relabel.
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2
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31
2
31
3
2
12
5
5
Select, then relabel/push
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21
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1
s
2
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t
Select an active node.
0
5
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2
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t
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5s202
2
1
12
26
s
s
Push/Relabel
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2
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31
3
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22
1
41
Select, then relabel/push
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21
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1
s
2
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5
t
Select an active node.
0
5
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2
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t
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5s202
2
1
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26
s
s
There is no incident admissible arc. So relabel.
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2
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21
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2
31
3
2
22
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41
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5
5
Select, then relabel/push
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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
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
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2
3
21
3
22
222
41
3
5
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3
1
141
4
Select, then relabel/push
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11
4
21
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3
1
s
2
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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
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3
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222
41
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5
5
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1
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2
Select, then relabel/push
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s
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t
Select an active node.
0
5
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2
1
t
4 5
s02
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1
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26
s
s
Push/Relabel
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2
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222
41
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1
4
4
2
4
Select, then relabel/push
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21
2
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1
s
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5
t
Select an active node.
0
5
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2
1
t
4 5
s02
2
1
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26
s
s
Push/Relabel
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2
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21
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41
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5
Select, then relabel/push
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s
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t
One can keep pushing flow between nodes 2 and 5 until eventually all flow returns to node s.
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t
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s02
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s
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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Assignments
Read sections 7.3 and 7.6
Exercises
7.4
