Network Optimization
Flow Decomposition
2
The initial flow
1
2 4
53
6
2
3
47
3
4 48
59
A deficit node (more flow leaving than entering).An excess node (more flow entering than leaving)
A balanced node (flow in = flow out)
2 -2
3
Find a Path or Cycle W
1
2 4
53
6
2
3
47
3
4 48
59
2 -21
Carry out a depth first search. Stop when a node with excess is reached or when there is a cycle.
Select a node with deficit if there is one.
Determine the capacity of the walk W.
The capacity of 1-2-4-5-3-1 is 2.
2 4
53
1
4
Updates
1
2 4
53
6
2
3
47
3
4 48
59
2 -21
Subtract the flow in W from the current flow.
Add the flow in W to the decomposition.
0
25
6
7
cycle flows path flows
2 units around 1-2-4-5-3-1
5
Find a path or cycle W
1
2 4
53
63
25
3
4 46
57
2 -21Carry out a dfs.
Select a node with deficit if there is one.
Determine the capacity of W.
2 4
53
The capacity of 2-4-5-3-2 is 3.
6
Updates
Add the cycle flow to the decomposition
update the current flow
cycle flows path flows
2 units around 1-2-4-5-3-13 units around 2-4-5-3-2
1 63
25
3
4 46
57
-212 0
2
3
4
2 4
53
7
Find a path or cycle using dfs
1 6
22
3
4 43
54
-212
2 4
53
Carry out a dfs.
Select a node with deficit if there is one.
Determine the capacity of W.
6
22
3
-212
2 4
capacity of a path = min {arc capacity, excess, deficit} = 2
8
Updates
22
3
4 43
54
-22
53
Add the path flow to the decomposition
update the current flow
cycle flows path flows
2 units around 1-2-4-5-3-13 units around 2-4-5-3-2
2 units in 1-2-4-6
00
1
00 1 6
2 4
1 6
9
Find a path or cycle using dfs
1
4 43
5453
1 6
2 4
1 6
Carry out a dfs.
Select a node with deficit if there is one. Otherwise, select any node with flow leaving.
Determine the capacity of W.
3
5
4
6
3
The capacity is 1
10
Updates
1
4 43
5453
1 6
2 4
1 6
Add the cycle flow to the decomposition
update the current flow
2 units around 1-2-4-5-3-13 units around 2-4-5-3-2
2 units in 1-2-4-6
1 unit around 3-4-6-5-3
0
3
43
cycle flows path flows
11
Find a path or cycle using dfs
3 43
4353
1 6
2 4
1 6
Carry out a dfs.
Select a node with deficit if there is one. Otherwise, select any node with flow leaving.
Determine the capacity of W.
3
The capacity of 3-4-5-3 is 3
12
Updates
3 43
4353
1 6
2 4
1 6
Add the cycle flow to the decomposition
update the current flow
2 units around 1-2-4-5-3-13 units around 2-4-5-3-2
2 units in 1-2-4-61 unit around 3-4-6-5-33 units around 3-4-5-3
cycle flows path flows
13
Find a path or cycle using dfs
4
453
1 6
2 4
1 6
Carry out a dfs.
Select a node with deficit if there is one. Otherwise, select any node with flow leaving.
Determine the capacity of W.
5
14
Updates and the final flow decomposition
53
1 6
2 4
1 64
4
2 units around 1-2-4-5-3-13 units around 2-4-5-3-2 2 units in 1-2-4-61 unit around 3-4-6-5-33 units around 3-4-5-3
Add the cycle flow to the decomposition
update the current flow
4 units around 5-6-5
cycle flows path flows
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15.082J / 6.855J / ESD.78J Network OptimizationFall 2010
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