Network Optimization Flow Decomposition.

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Presentation transcript:

Network Optimization Flow Decomposition

The initial flow 7 2 4 4 3 3 8 2 -2 1 6 4 4 2 3 5 5 9 A deficit node (more flow leaving than entering). An excess node (more flow entering than leaving) A balanced node (flow in = flow out)

Find a Path or Cycle W Select a node with deficit if there is one. 7 2 4 4 3 3 8 2 1 1 6 -2 4 4 2 Carry out a depth first search. Stop when a node with excess is reached or when there is a cycle. 3 5 5 9 2 4 5 3 1 Determine the capacity of the walk W. The capacity of 1-2-4-5-3-1 is 2.

Updates Add the flow in W to the decomposition. 7 5 2 4 4 2 3 3 6 8 2 1 1 6 -2 4 4 Subtract the flow in W from the current flow. 2 3 5 5 9 7 2 units around 1-2-4-5-3-1 cycle flows path flows

Find a path or cycle W Select a node with deficit if there is one. 5 2 4 2 3 3 6 2 1 1 6 -2 4 4 Carry out a dfs. Determine the capacity of W. 3 5 5 7 2 4 5 3 The capacity of 2-4-5-3-2 is 3.

Updates Add the cycle flow to the decomposition 2 5 2 4 2 3 3 6 3 2 1 6 3 2 1 1 6 -2 4 4 update the current flow 3 5 5 4 7 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 cycle flows path flows

Find a path or cycle using dfs Select a node with deficit if there is one. 2 2 4 2 3 3 2 1 1 6 -2 4 4 Carry out a dfs. Determine the capacity of W. 3 5 5 4 6 2 3 -2 1 4 capacity of a path = min {arc capacity, excess, deficit} = 2

Updates Add the path flow to the decomposition 2 2 4 2 1 3 3 2 1 1 6 6 2 2 4 2 1 3 3 2 1 1 6 6 -2 4 4 update the current flow 3 5 5 4 2 units around 1-2-4-5-3-1 2 units in 1-2-4-6 3 units around 2-4-5-3-2 cycle flows path flows

Find a path or cycle using dfs Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 1 3 1 1 6 6 4 4 3 3 5 5 4 Carry out a dfs. 5 4 6 3 Determine the capacity of W. The capacity is 1

Updates Add the cycle flow to the decomposition 2 4 1 3 1 1 6 6 3 4 4 3 1 1 6 6 3 4 4 update the current flow 3 5 4 5 4 3 2 units around 1-2-4-5-3-1 2 units in 1-2-4-6 3 units around 2-4-5-3-2 1 unit around 3-4-6-5-3 cycle flows path flows

Find a path or cycle using dfs Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 3 1 1 6 6 3 4 3 3 5 4 3 Carry out a dfs. Determine the capacity of W. The capacity of 3-4-5-3 is 3

Updates Add the cycle flow to the decomposition 2 4 3 1 1 6 6 3 4 update the current flow 3 5 4 3 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 2 units in 1-2-4-6 1 unit around 3-4-6-5-3 3 units around 3-4-5-3 cycle flows path flows

Find a path or cycle using dfs Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 1 1 6 6 4 3 5 5 4 Carry out a dfs. Determine the capacity of W.

Updates and the final flow decomposition Add the cycle flow to the decomposition 2 4 1 1 6 6 4 update the current flow 3 5 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 2 units in 1-2-4-6 1 unit around 3-4-6-5-3 3 units around 3-4-5-3 4 units around 5-6-5 cycle flows path flows

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