- The max. flow algorithm consists
of 2 parts:
- A labelling procedure:
- Label the appropriate vertices in order to find a flow augmenting path from S → T
- A flow augmentation procedure:
- Increases the flow along the flow augmenting path found in the previous step.
- A labelling procedure:
- Psuedo code:
for ( each edge e in network ) do { f(e) = 0; } repeat { Clear the label of all nodes; Use DFS to find a path S → T { During DFS traversal: 1. Use only "flow augmenting edges" (1. forward edges that are unsaturated or 2. backward edges with flow > 0) 2. if ( node visited is unlabeled ) label the node x with the predecessor node } if ( T received a label ) { // T was reached by a flow augmenting path !! 1. Use the labels to trace (back) the path; 2. increase flow by the slack of the (flow augmenting) path } } until T not reached during DFS
- Example: DFS with
labeling
- Sample flow:
- We start in
S.
The only flow augmenting edge available is: S → C.
DFS visits and labels the node C:
- The DFS continues from node C
The only flow augmenting edge available is: C → E.
DFS visits and labels the node E:
- The DFS continues from node E
The only flow augmenting edge available is the backward edge: E → B !!!
DFS visits and labels the node B:
- The DFS continues from node B
The flow augmenting edges are:
- B → D
- B → F
(Edge B → E is disqualified because E is already labeled !!!)
Let's use B → D first....
DFS visits and labels the node D:
- The DFS continues from node D
The flow augmenting edges are:
- D → T
- The backward edge D → A
Let's use D → T first....
DFS visits and labels the node T:
DONE !!!
- Flow augmenting path:
Slack = 1
After increasing flow by 1:
- Sample flow:
- Basic network:
- Initial flow:
- Find a flow augmenting path using
DFS (traverse node in
alpha-numerical order)
(The algorithm uses labels.
The flow augmenting path is highlighted for your convenient.)
Augment flow: by 3
- Find a flow augmenting path using
DFS (traverse node in
alpha-numerical order)
Augment flow: by 1
- Find a flow augmenting path using
DFS (traverse node in
alpha-numerical order)
Augment flow: by 1
- Find a flow augmenting path using
DFS (traverse node in
alpha-numerical order)
Augment flow: by 1
- Find a flow augmenting path using
DFS (traverse node in
alpha-numerical order)
- No more flow augmenting paths
Done
- Find a maximum flow
in the following basic network:
Solution: (max flow = 17)
Flow augmenting paths are "color coded" with the given amount of flow passing through various links.