1
Chapter 26
Maximum Flow
How do we transport the maximum amount data from source to sink?
Some of these slides are adapted from
Lecture Notes of
Kevin Wayne.
Contents
±
Contents.
±
Maximum flow problem.
±
Minimum cut problem.
±
Maxflow mincut theorem.
±
Augmenting path algorithm.
±
Capacityscaling.
±
Shortest augmenting path.
±
An st cut is a node partition (S, T) s.t. s
∈
S, t
∈
T.
±
The capacity of an st cut (S, T) is:
±
Min st cut
:
find an st cut of minimum capacity.
Cuts
∑
∈
∀
∈
∈
V
v
u
T
v
S
u
v
u
c
,
,
).
,
(
s
2
3
4
5
6
7
t
15
5
30
15
10
8
15
9
6
10
10
10
15
4
4
Capacity = 30
Cuts
±
An st cut is a node partition (S, T) s.t. s
∈
S, t
∈
T.
±
The capacity of an st cut (S, T) is:
±
Min st cut:
find an st cut of minimum capacity.
s
2
3
4
5
6
7
t
15
5
30
15
10
8
15
9
6
10
10
10
15
4
4
Capacity = 62
∑
∈
∀
∈
∈
V
v
u
T
v
S
u
v
u
c
,
,
).
,
(
Cuts
±
An st cut is a node partition (S, T) s.t. s
∈
S, t
∈
T.
±
The capacity of an st cut (S, T) is:
±
Min st cut:
find an st cut of minimum capacity.
∑
∈
∀
∈
∈
V
v
u
T
v
S
u
v
u
c
,
,
).
,
(
s
2
3
4
5
6
7
t
15
5
30
15
10
8
15
9
6
10
10
10
15
4
4
Capacity = 28
Recall the example
s
2
3
4
5
6
7
t
15
5
30
15
10
8
15
9
6
10
10
10
15
4
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 Spring '09
 Flow network, Maximum flow problem, Maxflow mincut theorem, 2 Min

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