Download presentation

Presentation is loading. Please wait.

Published byTobias Masker Modified over 3 years ago

1
IEOR 4004 Maximum flow problems

2
Connectivity t t s s Q1: Can Alice send a message to Bob ? Yes if every (s,t)-cut contains at least one forward edge forward backward

3
Connectivity t t s s Q2: How fast? Send data in parallel Q1: Can Alice send a message to Bob ?

4
Edge capacity t t s s Idea: Data packets can share edges (bandwidth) 12341234 Two packets in parallel

5
t t s s 12341234

6
Conservation of flow t t s s 12341234 v net flow (excess) value of flow value of flow incoming outgoing

7
t t s s 12341234...

8
Forward paths do not suffice t t s s 12341234

9
Augmenting chain t t s s 12341234

10
Exponentially many steps t t s s

11
t t s s

12
t t s s Residual network t t s s Forward path Augmenting chain

13
Recall: Connectivity t t s s Q1: Is there a path from s to t? Yes if every (s,t)-cut contains at least one forward edge Else No forward backward

14
Flows and cuts t t s s 12341234 flow across a cut (forward flow – backward flow) value of a flow capacity of a cut (forward edges) flow across a cut ≤ capacity of the cut Weak duality

15
Maximum flow = Minimum cut t t 12341234 s s optimal solution Strong duality

16
Transportation problem Factories Retail stores Requirement for goods Production capacity... Can factories satisfy the demand of retail stores ? aiai aiai bjbj bjbj edge if i-th factory can deliver to j-th store t Maximum flow capacity production (capacity) demand (capacity) source target a1a1 a2a2 anan bmbm b1b1 b2b2 s

17
Transportation problem Factories Retail stores Requirement for goods Production capacity... Can factories satisfy the demand of retail stores ? t Maximum flow capacity limited production (capacity) limited demand (capacity) Units of flow 1 23 source target bmbm b1b1 b2b2 Example 1: n=m=3 a 1 =a 2 =a 3 =1 b 1 =b 2 =b 3 =1 Answer: Yes! a1a1 a2a2 anan s

18
Transportation problem Factories Retail stores Requirement for goods Production capacity Can factories satisfy the demand of retail stores ? t Maximum flow capacity production (capacity) demand (capacity) source target Example 2: n=m=3 a 1 =a 2 =1 a 3 =3 b 1 =3 b 2 =b 3 =1 Answer: No! Maximum flow = 4 < 5 3 rd factory does not deliver to 1 st retail store s Example 1: n=m=3 a 1 =a 2 =a 3 =1 b 1 =b 2 =b 3 =1

Similar presentations

OK

1 EE5900 Advanced Embedded System For Smart Infrastructure Static Scheduling.

1 EE5900 Advanced Embedded System For Smart Infrastructure Static Scheduling.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on animal kingdom class 9 Ppt on stress management for students Ppt on red ocean strategy with examples Ppt on channels of distribution examples Ppt on conservation of water resources Ppt on diode as rectifier Ppt online compressor sales Ppt on total station survey Ppt on active magnetic bearing Ppt on prefilled syringes