15'082 and 6'855J

1
15.082 and 6.855J

• Min Global Cut Animation

2
Initialize
Saturate the arcs out of node 1.
Update the residual network
3
Initialize
1
We will never push from node 1 again or into node
1.
4
6
5
3
2
Compute distances to node 2
Determine admissible arcs
4
Push/Relabel
1
1
4
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4
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4
1
3
6
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2
6
1
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1
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1
Select an active node
Carry out push/relabel
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Push/Relabel
1
1
4
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1
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1
Select an active node
Relabel. There are no admissible arcs.
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Push/Relabel
1
1
4
1
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4
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3
1
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2
1
1
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1
Select an active node
Push from node 3.
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Push/Relabel
1
1
4
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4
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1
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1
Select an active node
Relabel node 3. Rule no empty levels permitted.
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Gaps

• Let t denote the current sink node.
• Let dmax be the maximum distance label of a node
  other than the source node.
• An empty level is a value k with d(t) lt k lt dmax
  such that there is no node j with d(j) k.
• If we increase d(3) to 7, we create a gap. In
  such a case, we increase d(j) to dmax 1.

9
Push/Relabel
1
1
4
1
4
4
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3
3
6
5
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1
1
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1
2
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1
Relabel node 3 so no empty level is created.
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Cut Level
1
1
5
4
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4
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1
1
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1
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5
1
A cut level is a level with exactly one node that
needs to be relabeled.
There is no path in the residual network from a
node in a cut level to a node below it.
11
Cut-level rule

• Always select an active node that is below the
  lowest cut-level.
• If each active node is at or above a cut level,
  then stop with the max preflow/ min cut.

12
Push/Relabel
1
1
5
4
4
1
4
3
4
4
3
4
2
3
6
5
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1
1
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1
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0
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1
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5
1
Select an active node below the lowest cut level.
Push from node 4.
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Push/Relabel
1
5
4
1
4
3
3
5
3
4
2
3
6
5
2
1
2
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6
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1
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0
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1
2
5
1
Select an active node below the lowest cut level.
Push from node 6.
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Push/Relabel
1
5
4
1
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5
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3
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5
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1
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1
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5
1
Select an active node below the lowest cut level.
Push from node 5.
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Push/Relabel
1
5
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5
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1
5
Select an active node below the lowest cut level.
Relabel node 5 if it will not create an empty
level.
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End of finding the first cut
1
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5
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1
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1
There is no active node below the lowest cut
level.
The max preflow and min cut have been found.
Let T be all nodes that can reach the sink node.
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End of finding the first cut
1
1
5
4
1
4
3
5
4
3
4
1
2
3
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6
1
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0
2
1
2
5
Here is the first minimum cut.
18
Beginning of the second cut
1
5
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1
2
Choose a new sink node at the lowest level
Make node 2 a source node
Saturate arcs out of node 2.
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Beginning of the second cut
2
1
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1
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0
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2
We will not push from node 2 or into node 2
again.
There is no need to physically merge nodes 1 and
2.
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Push/Relabel
2
1
5
4
1
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3
3
5
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2
Select an active node
Relabel node 3 if it will not create an empty
level.
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Push/Relabel
2
1
5
4
1
4
3
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5
3
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1
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0
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2
Level 4 becomes a cut level
There is no active node below a cut level.
The second cut has been found.
Let T be all nodes that can reach the sink node.
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Push/Relabel
2
1
1
5
4
1
4
3
5
4
3
4
1
2
3
6
6
1
5
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3
0
1
2
5
Here is the 2nd cut.
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Starting the 3rd cut
2
1
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1
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2
Make node 5 a source node
Node 6 becomes the new sink node
Saturate arcs out of node 5.
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Starting the 3rd cut
2
5
1
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2
Level 3 is still a cut level
There are still no active nodes below a cut level
We have found the 3rd cut
Let T be all nodes that can reach the sink node.
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The 3rd cut
2
5
1
1
5
4
1
4
3
5
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3
4
1
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6
1
6
3
0
1
2
5
The above cut is the best cut with 1, 2, 5 on one
side and with node 6 on the other side.
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Starting the 4th cut
2
5
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Node 6 becomes a source node
Node 4 becomes the next source node
Saturate arcs out of node 6
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We have found the 4th and 5th cuts
2
5
6
1
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9
1
0
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2
The only arcs in the network are out of source
nodes
We can stop the algorithm and identify the
remaining cuts.
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Here are cuts 4 and 5
1
1
4
4
1
1
5
5
4
4
1
1
3
6
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1
1
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5
2
5
Cut 5
Cut 4
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The global min cut was cut number 2
The algorithm ends with the minimum cut with node
1 on the source side.