Min Global Cut Animation 15.082 and 6.855J Min Global Cut Animation Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

Initialize 1 3 2 5 6 4 Saturate the arcs out of node 1. Update the residual network

Initialize 1 1 3 2 5 6 4 5 4 3 2 1 We will never push from node 1 again or into node 1. 4 6 3 5 2 Compute distances to node 2 Determine admissible arcs

Push/Relabel 1 1 5 4 3 2 1 4 1 4 4 4 1 5 2 3 3 6 6 1 3 5 3 6 6 3 2 1 2 5 1 Select an active node Carry out push/relabel

Push/Relabel 1 1 5 4 3 2 1 4 1 4 4 4 3 1 5 2 3 3 6 6 1 3 5 6 6 6 2 1 2 5 1 Select an active node Relabel. There are no admissible arcs.

Push/Relabel 1 1 5 4 3 2 1 4 1 4 4 4 3 1 5 2 1 3 3 6 1 6 1 5 6 6 6 2 1 2 5 1 Select an active node Push from node 3.

Push/Relabel 1 1 5 4 3 2 1 4 1 4 4 4 3 5 1 2 3 3 6 1 6 2 5 6 6 6 2 1 2 5 1 Select an active node Relabel node 3. Rule: no empty levels permitted.

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.

Push/Relabel 1 1 5 4 3 2 1 4 1 3 4 4 4 3 5 1 2 3 3 6 1 6 2 5 6 6 6 2 1 2 5 1 Relabel node 3 so no empty level is created.

Cut Level 1 1 5 4 1 4 3 4 4 3 4 2 1 2 5 3 6 1 6 2 1 5 6 6 6 2 1 2 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.

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.

Push/Relabel 1 1 5 4 4 1 4 3 4 4 3 4 2 1 2 5 3 6 1 6 2 1 5 6 6 6 2 1 2 5 1 Select an active node below the lowest cut level. Push from node 4.

Push/Relabel 1 5 4 1 4 3 3 5 3 4 2 1 2 5 3 6 6 2 6 2 1 5 6 6 6 2 1 2 5 1 Select an active node below the lowest cut level. Push from node 6.

Push/Relabel 1 5 4 1 4 3 3 5 3 4 2 1 2 5 3 6 6 2 1 5 8 4 6 2 1 2 5 5 2 1 Select an active node below the lowest cut level. Push from node 5.

Push/Relabel 1 5 4 1 4 3 3 5 3 4 2 1 2 5 3 6 6 2 1 5 8 4 6 2 2 2 5 5 1 Select an active node below the lowest cut level. Relabel node 5 if it will not create an empty level.

End of finding the first cut 1 5 4 1 4 3 3 5 3 4 2 2 1 5 3 6 6 2 1 5 8 4 6 2 2 2 5 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.

End of finding the first cut 1 1 5 4 1 4 3 5 4 3 4 1 2 3 6 6 1 5 6 3 2 1 2 5 Here is the first minimum cut.

Beginning of the second cut 1 5 4 1 4 3 3 5 3 4 2 2 5 1 3 6 6 2 1 5 5 8 4 6 2 2 2 2 5 1 Choose a new sink node at the lowest level Make node 2 a source node Saturate arcs out of node 2.

Beginning of the second cut 1 2 5 4 1 4 3 3 5 3 4 2 5 7 2 3 6 6 2 1 5 8 4 2 2 5 5 We will not push from node 2 or into node 2 again. There is no need to physically merge nodes 1 and 2.

Push/Relabel 1 2 5 4 1 4 3 3 5 3 4 2 7 5 2 3 3 6 6 2 1 5 8 4 2 2 5 5 Select an active node Relabel node 3 if it will not create an empty level.

Push/Relabel 1 2 5 4 1 4 3 3 5 3 4 2 5 7 2 3 6 6 2 1 5 8 4 2 2 5 5 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.

Push/Relabel 1 2 1 5 4 1 4 3 5 4 3 4 1 2 3 6 6 1 5 6 3 1 2 5 Here is the 2nd cut.

Starting the 3rd cut 1 2 5 4 1 4 3 3 5 3 4 2 6 7 5 2 3 6 6 2 1 5 5 8 4 2 2 5 5 Make node 5 a source node Node 6 becomes the new sink node Saturate arcs out of node 5.

Starting the 3rd cut 1 2 5 5 4 1 4 3 3 5 3 4 2 5 7 2 3 6 6 2 1 2 2 5 5 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.

The 3rd cut 1 2 5 1 5 4 1 4 3 5 4 3 4 1 2 3 6 6 1 6 3 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.

Starting the 4th cut 1 2 5 5 4 4 1 4 3 3 5 3 4 2 6 5 7 2 3 6 6 2 1 2 2 5 5 Node 6 becomes a source node Node 4 becomes the next source node Saturate arcs out of node 6

We have found the 4th and 5th cuts 1 2 5 6 5 4 1 4 3 3 4 2 2 5 9 3 6 1 2 2 5 5 The only arcs in the network are out of source nodes We can stop the algorithm and identify the remaining cuts.

Here are cuts 4 and 5 1 1 4 4 1 1 5 5 4 4 1 1 3 6 3 6 6 6 3 3 1 1 2 5 2 5 Cut 5 Cut 4

The global min cut was cut number 2 5 4 3 2 1 6 The algorithm ends with the minimum cut with node 1 on the source side.