Download - 15.082 and 6.855J
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15.082 and 6.855J
Min Global Cut Animation
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Initialize
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Saturate the arcs out of node 1.
Update the residual network
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Initialize
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Compute distances to node 2
Determine admissible arcs
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We will never push from node 1 again or into node 1.
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Push/Relabel
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Select an active node
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Carry out push/relabel
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Push/Relabel
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Select an active node
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Relabel. There are no admissible arcs.
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Push/Relabel
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Select an active node
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Push from node 3.
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Push/Relabel
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Select an active node
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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) < k < 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.
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Push/Relabel
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Relabel node 3 so no empty level is created.
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Cut Level
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A cut level is a level with exactly one node that needs to be relabeled.
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There is no path in the residual network from a node in a cut level to a node below it.
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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.
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Push/Relabel
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Select an active node below the lowest cut level.
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Push from node 4.
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Push/Relabel
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Select an active node below the lowest cut level.
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Push from node 6.
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Push/Relabel
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Select an active node below the lowest cut level.
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Push from node 5.
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Push/Relabel
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Select an active node below the lowest cut level.
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Relabel node 5 if it will not create an empty level.
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End of finding the first cut
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There is no active node below the lowest cut level.
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The max preflow and min cut have been found.
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Let T be all nodes that can reach the sink node.
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End of finding the first cut
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Here is the first minimum cut.
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Beginning of the second cut
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Choose a new sink node at the lowest level
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Make node 2 a source node
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Saturate arcs out of node 2.
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Beginning of the second cut
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We will not push from node 2 or into node 2 again.
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There is no need to physically merge nodes 1 and 2.
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Push/Relabel
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Select an active node
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Relabel node 3 if it will not create an empty level.
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Push/Relabel
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Level 4 becomes a cut level
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There is no active node below a cut level.
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The second cut has been found.
Let T be all nodes that can reach the sink node.
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Push/Relabel
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Here is the 2nd cut.
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Starting the 3rd cut
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Make node 5 a source node
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Node 6 becomes the new sink node
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Saturate arcs out of node 5.
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Starting the 3rd cut
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Level 3 is still a cut level
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There are still no active nodes below a cut level
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We have found the 3rd cut
Let T be all nodes that can reach the sink node.
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The 3rd cut
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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
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Node 6 becomes a source node
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Node 4 becomes the next source node
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Saturate arcs out of node 6
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We have found the 4th and 5th cuts
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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
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Cut 4 Cut 5
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The global min cut was cut number 2
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The algorithm ends with the minimum cut with node 1 on the source side.
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