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
![Page 1: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/1.jpg)
ENGM 631
Maximum Flow Solutions
![Page 2: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/2.jpg)
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)
![Page 3: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/3.jpg)
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]
![Page 4: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/4.jpg)
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'
![Page 5: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/5.jpg)
Maximum Flow
1. Find an augmenting path2. Determine the maximum flow augmentation
possible3. Augment flow by that amount
![Page 6: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/6.jpg)
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
![Page 7: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/7.jpg)
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)
![Page 8: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/8.jpg)
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)
![Page 9: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/9.jpg)
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)
![Page 10: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/10.jpg)
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)
![Page 11: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/11.jpg)
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)
![Page 12: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/12.jpg)
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)
![Page 13: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/13.jpg)
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)
![Page 14: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/14.jpg)
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)
![Page 15: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/15.jpg)
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)
![Page 16: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/16.jpg)
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)
![Page 17: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/17.jpg)
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)
![Page 18: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/18.jpg)
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
![Page 19: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/19.jpg)
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)
![Page 20: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/20.jpg)
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
![Page 21: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/21.jpg)
(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
![Page 22: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/22.jpg)
(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
![Page 23: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/23.jpg)
(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
![Page 24: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/24.jpg)
(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
![Page 25: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/25.jpg)
(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
![Page 26: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/26.jpg)
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)
![Page 27: ENGM 631 Maximum Flow Solutions. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6)…](https://reader035.vdocuments.site/reader035/viewer/2022062414/5a4d1c0f7f8b9ab0599f5b0c/html5/thumbnails/27.jpg)
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)