1. CSC 172 DATA STRUCTURES

2. FLOW

3. FLOW
“I can go with the flow.”
- Queens of the Stone Age

4. “Let your body go with the flow.”
“I can go with the flow.”
- Queens of the Stone Age
“Let your body go with the flow.”
- Madonna

5. “Let your body go with the flow.”
“I can go with the flow.”
- Queens of the Stone Age
“Let your body go with the flow.”
- Madonna
“Use the flow, Luke!”
- Obi-wan Kenobi

6. FLOW NETWORKS Three categories: Distribution problems
Matching problems
Cut problems

7. FLOW NETWORKS Distribution problems
-moving objects from one place to another in a network
Merchandise distribution
Communication
Traffic flow 1 2 3 4 5 6 7 8 9

8. FLOW NETWORKS Matching problems -connecting pairs Job Placement
Minimum distance point matching 1 1 2 2 3 3 4 4 A B C D E

9. MATCHING JOB ASSINGMENTS
Abe – Adobe, Apple, HP
Bea – Adobe, Apple, Yahoo
Cal – HP, IBM, Sun
Deb – Adobe, Apple
Eli – IBM, Sun, Yahoo
Fay – HP, Sun, Yahoo
Adobe – Abe, Bea, Deb
Apple – Abe, Bea, Deb
HP – Abe, Cal, Fay
IBM – Cal, Eli
Sun – Cal, Eli, Fay
Yahoo – Bea, Eli, Fay

10. FLOW NETWORKS Cut problems
-remove edges to break graph into two pieces
Network reliability
Cutting supply lines 1 1 2 2 3 3 4 4 5

11. FLOW NETWORKS (DEF)
Flow Networks :

12. FLOW NETWORKS (DEF) s
Flow Networks :
Digraphs A B C D E t

13. FLOW NETWORKS (DEF) Flow Networks : Digraphs
Weights on edges (capacities) 2 2 1 A B 2 2 1 2 C 2 4 D E 2 1 1 t

14. FLOW NETWORKS (DEF) Flow Networks : Digraphs
source s
Flow Networks :
Digraphs
Weights on edges (capacities)
Two special verticies
Source ; s
Sink : t 2 2 1 A B 2 2 1 2 C 2 4 D E 1 2 1 t
sink

15. CAPACITY AND FLOW Edge Capacity non-negative weights t s 2 2 1 A B 2 24 D E 1 2 1 1 t

16. CAPACITY AND FLOW Edge Capacity non-negative weights Flow
0 <= flow <= capacity
flow in = flow out 2 2 1 2 1 A B 2 2 1 1 1 1 1 2 C 2 4 D 1 E 1 2 1 1 2 1 t

17. CAPACITY AND FLOW Edge Capacity non-negative weights Flow
value = 3 s
Edge Capacity
non-negative weights
Flow
0 <= flow <= capacity
flow in = flow out
Value
outflow of source
inflow of sink 2 2 1 2 1 A B 2 2 1 1 1 1 1 2 C 2 4 D 1 E 1 2 1 1 2 1 t
value = 3

18. RESIDUAL GRAPH Residual = capacity - flow t s 2 2 1 2 1 A B 2 2 1 1 1
value = 3 s
Residual =
capacity - flow 2 2 1 2 1 A B 2 2 1 1 1 1 1 2 C 2 4 D 1 E 1 2 1 1 2 1 t
value = 3

19. RESIDUAL CAPACITY Residual = capacity - flow t s 2 2 1 1 1 2 1 A B 1 2
value = 3 s
Residual =
capacity - flow 2 2 1 1 1 2 1 A B 1 2 2 1 1 1 1 1 1 2 C 1 2 4 1 4 D 1 E 1 2 1 1 1 2 1 t
value = 3

20. RESIDUAL GRAPH Residual = capacity - flow t s 1 A 1 B 1 1 1 C 1 D 4 E

21. AUGMENTATION PATH Augmentation Paths
Augmentation Path
- a path from source to sink through the residual graph 1 A 1 B 1 1 1 C 1 D 4 E 1 t

22. BACKEDGE For every edge, there is a virtual reverse back edge
the capacity of a back edge is the flow of the edge 2 2 1 2 1 A B 2 2 1 1 1 1 2 1 C 2 4 D 1 E 1 2 1 1 2 1 t

23. BACKEDGE For every edge, there is a virtual reverse back edge
the capacity of a back edge is the flow of the edge (whoa!)
Backedge(v,w) = flow(w,v) 2 2 2 1 1 2 1 A B 1 2 2 1 1 1 1 1 2 1 C 1 2 4 1 D 1 E 1 2 1 1 2 1 2 1 t

24. BACKEDGE GRAPH For every edge, there is a virtual reverse back edge
the capacity of a back edge is the flow of the edge (whoa!)
Backedge(v,w) = flow(w,v) 2 1 A B 1 1 1 C 1 1 D E 1 2 t

25. RESIDUAL GRAPH Residual(w,v) = capacity(w,v) – flow(w,v)
Backedge(v,w) = flow(w,v) 2 1 1 A A 1 1 B B 1 1 1 1 1 C C 1 1 D D 1 4 E E 1 1 2 t t

26. Of course, in real life : There is one graph public class Edge {
int v, w ;
int capacity, flow;
int residual, backflow;
// methods }

27. Define Capacity Graph Flow graph
residual graph (capacity – flow at each edge)
back edge graph – reverse edge directions
augmenting path – from s to t along non-zero capacity edges

28. Why all the complexity?
Sometimes, in graph theory as in life, it turns out that greed leads us to sub-optimal results.

29. s 3 2 1 A B 3 4 2 D E 2 3 t

30. s 3 2 1 A B 3 4 2 D E 2 3 t

31. s 3 2 3 1 A B 3 4 2 3 D E 2 3 3 t

32. s 2 1 A B 3 1 2 D E 2 t

33. capacity, flow, residual, backedge3 2 3 2 1 1 A B A B 4 3 2 3 2 4 D E D E 2 3 3 2 t t

34. capacity, flow, residual, backedge3 2 3 2 3 1 1 A B A B 4 3 2 3 2 4 3 D E D E 2 3 3 3 2 t t

35. capacity, flow, residual, backedge3 2 2 3 3 1 1 A B A B 1 3 2 3 2 4 3 3 D E D E 2 3 3 2 3 t t

36. capacity, flow, residual, backedge3 2 2 3 3 1 1 A B A B 1 3 2 3 2 4 3 3 D E D E 2 3 3 2 3 t t

37. capacity, flow, residual, backedge3 2 2 3 2 3 1 1 A B A B 1 3 2 2 2 3 2 4 1 3 D E D E 2 3 2 3 2 3 t t

38. capacity, flow, residual, backedge3 2 3 2 3 1 2 1 A B A B 3 3 2 2 2 3 2 2 4 1 1 D E D E 2 3 2 2 3 3 t t

39. capacity, flow, residual, backedge3 2 3 2 3 1 2 1 A B A B 3 3 2 2 2 3 2 2 4 1 1 D E D E 2 3 2 2 3 3 t t