Flow network

1. Lecture 13
Introduction to Maximum Flows

2. Flow Network





















V
y
V
y
f
v
y
s
f
f
t
s
V
x
y
x
f
V
y
x
x
y
f
y
x
f
V
y
x
y
x
u
y
x
f
R
V
V
f
G
t
s
y
x
u
E
y
x
y
x
u
E
y
x
E
V
G
))
(
(
)
,
(
|
|
}.
,
{
all
for
0
)
,
(
on)
conservati
(Flow
,
,
all
for
)
,
(
)
,
(
symmetry)
(Skew
,
,
all
for
)
,
(
)
,
(
)
constraint
(Capacity
such that
:
function
a
is
in
flow
A
.
sink
a
and
source
a
:
nodes
h two
distinguis
We
.
0
)
,
(
assume
,
)
,
(
For
.
0
)
,
(
capacity
e
nonnegativ
a
ha
)
,
(
edge
each
in which
graph
directed
a
is
)
,
(
network
flow
A

3. s t
out
flow
in
flow 
|
|
out
flow
f |
|
in
flow
f
 
 


V
y V
y
t
y
f
y
s
f
f )
,
(
)
,
(
|
|

4. 4

8. The Ford Fulkerson Maximum Flow Algorithm
• Begin x := 0;
– create the residual network G(x);
– while there is some directed path from s to t in G(x)
do
• begin
• let P be a path from s to t in G(x);
• ∆:= δ(P);
• send ∆ units of flow along P;
• update the r's;
– End
• end {the flow x is now maximum}.

9. The Ford-Fulkerson Augmenting Path
Algorithm for the Maximum Flow Problem

10. Ford-Fulkerson Max Flow
4
1
1
2
2
1
2
3
3
1
s
2
4
5
3
t
This is the original network,
and the original residual
network.

11. 4
1
1
2
2
1
2
3
3
1
Ford-Fulkerson Max Flow
Find any s-t path in G(x)
s
2
4
5
3
t

12. 4
1
1
2
1
3
Ford-Fulkerson Max Flow
Determine the capacity D of the path.
Send D units of flow in the path.
Update residual capacities.
1
1
1
2
1
2
3
2
1
s
2
4
5
3
t

13. 4
1
1
2
1
3
Ford-Fulkerson Max Flow
Find any s-t path
1
1
1
2
1
2
3
2
1
s
2
4
5
3
t

14. 4
2
1
1
1
1
2
2
1
1
1
1
3
Ford-Fulkerson Max Flow
1
1
1
1
3
2
1
s
2
4
5
3
t
Determine the capacity D of the path.
Send D units of flow in the path.
Update residual capacities.

15. 4
2
1
1
1
1
2
2
1
1
1
1
3
Ford-Fulkerson Max Flow
1
1
1
1
3
2
1
s
2
4
5
3
t
Find any s-t path

16. 1
1 1
1
1
4
1
2
1
1
2
1
1
3
Ford-Fulkerson Max Flow
1
1
3
2
1
s
2
4
5
3
t
Determine the capacity D of the path.
Send D units of flow in the path.
Update residual capacities.

17. 1
1 1
1
1
4
1
2
1
1
2
2
1
1
3
Ford-Fulkerson Max Flow
1
1
3
2
1
s
2
4
5
3
t
Find any s-t path

18. 1
1
1
2 1 1
1
1
4
2
2
1
1
2
2
1
Ford-Fulkerson Max Flow
1
1
3
1
1
s
2
4
5
3
t
Determine the capacity D of the path.
Send D units of flow in the path.
Update residual capacities.
2

19. 1
1
2 1 1
1
1
4
2
2
1
1
2
2
1
Ford-Fulkerson Max Flow
1
1
3
1
1
s
2
4
5
3
t
Find any s-t path
2

20. 1
1
1
1
1
4
1
3
1
1
2 1 1
3
2
2
1
2
1
Ford-Fulkerson Max Flow
2
1
s
2
4
5
3
t
2
Determine the capacity D of the path.
Send D units of flow in the path.
Update residual capacities.

21. 1
1
1
1
1
4
1
3
1
1
2 1 1
3
2
2
1
2
1
Ford-Fulkerson Max Flow
2
1
s
2
4
5
3
t
2
There is no s-t path in the residual
network. This flow is optimal

22. 1
1
1
1
1
4
1
3
1
1
2 1 1
3
2
2
1
2
1
Ford-Fulkerson Max Flow
2
1
s
2
4
5
3
t
2
These are the nodes that are reachable
from node s.
s
2
4
5
3

23. Ford-Fulkerson Max Flow
1
1
2
2
2
1
2
s
2
4
5
3
t
Here is the optimal flow

28. Review for cutset theorem

30. s t
S T
cut
-
of
capacity
)
to
from
(flow
)
to
from
flow
(
)
to
from
flow
(
|
|
value
flow
t
s
T
S
S
T
T
S
f





34. s
s
S
t
t
S
T
T
x
G
G
i j
i
j
k
k
l
l
ij
ij u
x 
0

kl
x
0
in
exist
not
does
)
,
(
in
exist
not
does
)
,
(




kl
x
ij
ij
x
x
G
k
l
u
x
G
j
i

35. )
,
(
cut
of
capacity
to
from
flow
),
,
( T
S
T
S
u
x
j
i ij
ij 



0
to
from
flow
0
),
,
( 


 S
T
x
l
k kl
)
,
(
of
capacity
)
to
from
(flow
)
to
from
(flow
|
|
value
flow
T
S
S
T
T
S
x




37. Quiz Sample
time?
-
polynomial
in
runs
algorithm
Fulkersor
-
Ford
Does
No!
:
Answer

44. Polynomial-time
 
 
digits.
4
has
10)
(i.e.,
1010
number
binary
The
.
4
10
log
e.g.,
.
log
is
integer
an
of
size
input
The
size.
input
its
to
respect
ith
function w
a
as
polynomial
a
by
bounded
is
time
running
its
if
time
-
polynomial
is
algorithm
An
2
2

m
m