1. CS 312: Algorithm Analysis Lecture #27: Network Flow This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License Slides by: Eric Ringger, with contributions from Mike Jones

2. Announcements  Project #5: Gene Sequence Alignment  Due: today  Homework #19, 20 due  HW 20 problem 3 moved to Monday  For you, what was the hardest step in formulating the solution to the longest common sub-sequence problem?  Project #6: Linear Programming  Help session: maybe Monday  HW #21: problem formulation for Proj. #6  Due Monday

3. Objectives  Review Standard Form (after step #2)  Consider the Network Flow problem  Work through a Simplex-like algorithm for maximizing flow  Understand the intuition behind “slack”

4. Standard Form Step #2  Introduce slack variable s i to produce an equality constraint from the i-th  -constraint:  What are these slack variables?  What do they represent?  What is their purpose?  We’ll get some intuition by examining network flow …

5. Network Flow Problem  Given a weighted DAG.  G = {V, E}  Weights represent max capacity of an edge.  s = source, t = sink  V  Goal: Maximize flow from s to t  Why limited to only one source and one sink? s b d t 46 21 3 We can transform a multiple source problem into a single source problem by adding a “super” source that feeds into the old sources with large weights. Ditto for sinks

6. Network Flow as LP s b d t 46 21 3

8. s b d t 46 21 3 What is the LP problem for this Network Flow Problem? Objective: Constraints:

9. 1.Choose a path. 2.Increase flow as much as possible on that path 3.Repeat until no paths remain Toward the Simplex Algorithm for Max Flow s b d t 46 21 3 Let’s try a simple greedy algorithm:

10. Toward the Simplex Algorithm for Max Flow 1.Choose a path. 2.Increase flow as much as possible on that path 3.Repeat until no paths remain Pick s,b,t first s b d t 46 21 3

11. 1.Choose a path. 2.Increase flow as much as possible on that path 3.Repeat until no paths remain Toward the Simplex Algorithm for Max Flow Could we have tried another path to begin with? s b d t 46 21 3

12. Toward the Simplex Algorithm for Max Flow 1.Choose a path. 2.Increase flow as much as possible on that path 3.Repeat until no paths remain What if we’d chosen s,d,t first? s b d t 46 21 3

13. Toward the Simplex Algorithm for Max Flow s b d t 4 6 2 1 3 1.Choose a path. 2.Increase flow as much as possible on that path 3.Repeat until no paths remain What if we’d chosen s,b,d,t first?

14. Moral of the Story  The above naïve greedy algorithm can lead to sub-optimal results.  We would still like to be able to pick any path without any commitment to analyze paths carefully.  But we require an algorithm giving optimal flow.  We need a way to give back or undo the blocking flow (e.g., from b to d) -- without “back-tracking” s b d t 4 of 43 of 6 0 of 21 of 1 1 of 3

15. Max Flow Algorithm s b d t 1 of 46 0 of 2 1 of 1 1 of 3 What we hadWhat we want s b d t 1 of 41 of 6 1 of 2 1 of 1 1 of 3 1

16. In the Original Graph  What really happens in the original graph:  flow on (s,b) is re-routed to (b,t) instead of through b,d,t  flow from (s,d) is sent along (d,t),  but we don’t have to think about re-routing  we’re just allocating flow in an updated graph without consideration of what we did before. s b d t 1 of 41 of 6 1 of 2 1 of 1 1 of 3 1

17. Simplex for Max Flow s b d t 1 of 40 of 6 1 of 1 1 of 3 Network after maximizing s,b,d,t 0 of 2

18. Simplex for Max Flow Network after maximizing s,b,d,t Residual (or slack) network s b d t 3 6 2 2 1 0 1 s b d t 1 of 40 of 6 1 of 1 1 of 3 0 of 2

19. Simplex for Max Flow s b d t 1 of 40 of 6 1 of 1 1 of 3 Network after maximizing s,b,d,t 0 of 2 Residual (or slack) network s b d t 3 6 2 2 1 0 Do: Choose new simple path with non-0 capacity on every edge (How?) Take up slack (i.e., residual capacity) along the new path Transform the problem from G to G R Until we find no more slack (i.e., residual capacity) in the network. 1

20. Simplex for Max Flow s b d t 3 6 2 2 1 0 1

21. Flow Schedule  Book-keeping:  accumulate flow along each original edge  OR:  just check the reverse edges!

22. Max Flow Algorithm

23. Status  We have a working version of a simplex-like algorithm for Max Flow  Take up slack  Transform the problem  Iterate  Simplex has these basic steps as well  Simplex will focus on one constraint at a time.  Next time: more examples

24. Assignment  HW #20  Due: Friday  Problem #7.10 is a network flow problem  Read about “min cut” in order to be able to answer the whole question.

25. Example: Optimum Bandwidth Allocation  Communications network between three users A,B, and C with bandwidths as shown:

26.  Variables:  x AB = short path bandwidth allocated to the connection between A & B  x AB ’ = long path bandwidth allocated to the connection between A & B  Likewise for BC  And AC Example: Optimum Bandwidth Allocation

27.  Objective:  Constraints on “last mile” network bandwidth:  Constraints on Inner network bandwidth:  Constraints on Customer bandwidth:  Non-negativity constraints: Example: Optimum Bandwidth Allocation

28.  Solution (via Simplex) Example: Optimum Bandwidth Allocation

29. Issues:  One variable for every possible path between the users.  What’s the problem?