engm 631 maximum flow solutions. maximum flow models (flow, capacity) (0,3) (2,2) (5,7) (0,8)...
DESCRIPTION
Maximum Flow Models (Flow, Capacity) [External Flow] (0,3) (2,2) (5,7) (0,8) (3,6) (6,8) (3,3) (4,4) (4,10) Maximal Flow 1.Capacity is only relevant parameter. 2.Find maximal flow from source to sink. S S [M] [-M]TRANSCRIPT
ENGM 631
Maximum Flow Solutions
Maximum Flow Models(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Maximum Flow Models(Flow, Capacity)[External Flow]
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Maximal Flow1. Capacity is only relevant
parameter.2. Find maximal flow from
source to sink.
S S
[M] [-M]
Maximum Flow 1. Find a flow augmenting path defined by a sequence of arcs P
=(k1, k2,.v.v.vkp) 2. Determine the maximum flow increase along the path
3. Change the flow in the arcs on the path
4. Repeat until no flow augmenting paths can be found
),:(minmin
),:min(
Pkkx
Pkkxu
k
kk
0
0
kk xx'
Maximum Flow
1. Find an augmenting path2. Determine the maximum flow augmentation
possible3. Augment flow by that amount
Maximum Flow Models(Flow, Capacity)
(0,3
)
(0,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(0,8)
(0,3)
(0,4)
(0,10)
Find a path top to bottom that has Additional capacity. Increase flow to Available capacity
Augmented Path(Flow, Capacity)
(0,3
)
(0,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(4,8)
(0,3)
(4,4)
(4,10)
(4) (4)
Augmented Path(Flow, Capacity)
(0,3
)
(0,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(4,8)
(0,3)
(4,4)
(4,10)
(4) (4)
Augmented Path(Flow, Capacity)
(0,3
)
(0,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(4,8)
(0,3)
(4,4)
(4,10)
(4) (6)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(6,8)
(0,3)
(4,4)
(4,10)
(6) (4)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(6,8)
(0,3)
(4,4)
(4,10)
(6) (4)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(6,8)
(0,3)
(4,4)
(4,10)
(6) (4)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(0,7)
(0,8
)
(0,6)
1
2
3
4
5
6
(6,8)
(0,3)
(4,4)
(4,10)
(6) (4)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(3,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(9) (9)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(3,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Arc 2-4 at capacity
(9) (9)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(3,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Arc 2-4 at capacityArc 2-5 at capacity
(9) (9)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(3,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Arc 2-4 at capacityArc 2-5 at capacityArc 3-5 at capacity
(9) (9)
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(3,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(9) (9)
No other path exists start to end that has additional capacity
Augmented Path(Flow, Capacity)
(0,3
)
(2,2)(3,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(9) (9)
Minimum Cut Algorithm
1. Find all possible cuts source to sink 2. Find the cut that has minimal capacity 3. Minimal capacity cut = maximum flow
(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 14)
Minimum Cut Algorithm
(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 11)
Minimum Cut Algorithm
(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 11)
Minimum Cut Algorithm
(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 11)
Minimum Cut Algorithm
(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 17)
Minimum Cut Algorithm
Maximum Flow Models(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 9)
Maximum Flow Models(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
(Capacity = 9)