15.082 and 6.855j the capacity scaling algorithm
TRANSCRIPT
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15.082 and 6.855J
The Capacity Scaling Algorithm
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The Original Costs and Node Potentials
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3 5
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4
12
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0
0 0
00
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The Original Capacities and Supplies/Demands
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3 5
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10
2020
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25
20
30
23
5 -2
-7 -19
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Set = 16. This begins the -scaling phase.
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3 5
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5 -2
-7 -19
We send flow from nodes with excess to nodes with deficit .
We ignore arcs with capacity .
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Select a supply node and find the shortest paths
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12
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5
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0
6 8
8shortest path distance
The shortest path tree is marked in bold and blue.
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Update the Node Potentials and the Reduced Costs
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3 5
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12
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5
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0
-7 -8
-8-6
0
0
0
06
3
1
To update a node potential, subtract the shortest path distance.
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Send Flow Along a Shortest Path in G(x, 16)
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Send flow from node 1 to node 5.
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5 -2
-7 -19
How much flow should be sent?
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Update the Residual Network
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19 units of flow were sent from node 1 to node 5.
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25
1
3023
5 -2
-70
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-19
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19
19
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This ends the 16-scaling phase.
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The -scaling phase continues whene(i) for some i.
e(j) - for some j.
There is a path from i to j.
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5 -2
-7 0
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19
19
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This begins and ends the 8-scaling phase.
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3 5
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The -scaling phase continues whene(i) for some i.
e(j) - for some j.
There is a path from i to j.
2020
6
25
1
30
5 -2
-7 0
10
4
19
19
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This begins 4-scaling phase.
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1 2020
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25
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30
5 -2
-7 0
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19
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What would we do if there were arcs with capacity at least 4 and negative reduced cost?
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Select a “large excess” node and find shortest paths.
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3 5
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-7 -8
-8-6
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0
0
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3
0
0 The shortest path tree is marked in bold and blue.0
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Update the Node Potentials and the Reduced Costs
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3 5
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1 00
-7 -8
-8-6
0
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0
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0
0
1
-11
-12-10
-4
To update a node potential, subtract the shortest path distance.
Note: low capacity arcs may have a negative reduced cost
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Send Flow Along a Shortest Path in G(x, 4).
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3 5
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1 2020
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5 -2
-7 0
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Send flow from node 1 to node 7
How much flow should be sent?
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Update the Residual Network
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3 5
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1 1620
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5 -2
-3 0
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4 units of flow were sent from node 1 to node 3
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-7
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This ends the 4-scaling phase.
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1 1620
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5 -2
-3 0
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There is no node j with e(j) -4.
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Begin the 2-scaling phase
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5 -2
-3 0
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There is no node j with e(j) -4.
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4 What would we do if there were arcs with capacity at least 4 and negative reduced cost?
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Send flow along a shortest path
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3 5
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1 1620
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1
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5 -2
-3 0
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15
0
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Send flow from node 2 to node 4
How much flow should be sent?
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Update the Residual Network
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3 5
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1 1620
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25
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5 -2
-3 0
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0
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2 units of flow were sent from node 2 to node 4
3 0
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Send Flow Along a Shortest Path
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3 5
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1 1620
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25
1
26
-3 0
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15
0
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Send flow from node 2 to node 3
3 0
How much flow should be sent?
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Update the Residual Network
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3 5
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1 1320
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-3 0
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0
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3 units of flow were sent from node 2 to node 3
3 0
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0
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This ends the 2-scaling phase.
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Are we optimal?
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0
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Begin the 1-scaling phase.
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Saturate any arc whose capacity is at least 1 and with negative reduced cost.
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reduced cost is negative
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Update the Residual Network
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-1
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Send flow from node 3 to node 1.
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1
0
Note: Node 1 is now a node with deficit
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Update the Residual Network
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1 unit of flow was sent from node 3 to node 1.
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Is this flow optimal?
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The Final Optimal Flow
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10,8
20,620
25,13
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20,20
30,3
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5 -2
-7 -19
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The Final Optimal Node Potentials and the Reduced Costs
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3 5
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00
-7 -11
-12-10
0
-4
0
12
0
Flow is at upper bound
Flow is at lower bound.