Max Flows

1
Max Flows

• ????? ?????????????????

2
?????? (flow)

• ????????????????????????????????????????????
• ??????? source s ??? sink t
• ??????????????????????????????????????? s ??? t

capacity
source
sink
????????????????? s ??? t ??????????????????
3
?????? (flow)

• ??? f(u,v) ??????????????????????????? (u,v)
• ?????????????????????????????????????????
• ??????????????????????? ??????????????????????????
  ????????? (???????? s ??? t)

flow / capacity
4
??????? (cut)
S
T
...
...
s
...
t
...
???????????????? (S,T) ????????????????????????
S ?? T
f(S,T) ? c(S,T)
12 4 11 ? 12 14
5
Max-Flow Min-Cut Theorem
S
T
...
...
s
...
t
...
fmax(S,T) cmin(S,T)
6
?????? max-flow

• Augmenting-path method
• Preflow-push method

7
??????????????????????
8
??????????????????????
9
Residual Network

• ???????????????? (residual capacity)
  ????????????? (u,v) ????????? G ??????????? f
  ??? cf(u,v) c(u,v) f(u,v)
• ????????????????? (residual network) ??? G
  ??????????? f ???
• ???????????????????? G
• ???????????? (u,v) ??????????? cf(u,v) ???
  cf(u,v) gt 0
• ???????????? (v,u) ??????????? f(u,v) ??? f(u,v)
  gt 0

10
Augmenting Path Method

• ?????????????? (augmenting path) ????????????????
  ?????????????????????????

12/12
11
Ford Fulkerson, 1962
12
????????
4
13
???????? (???)
14
???????? (???)
Residual Network
Resulting Flow
11
15
???????? (???)
16
???????? (???)
19
17
???????? (???)
18
???????? (???)
23
19
???????? (???)
20
?????????

• ????????????????????????????? ????????????????????
  ? ?????????????????????????????????
• ??? max flow ??? f, ?????????? O(E f)

21
??????? ?
22
Augmenting Path ????????

• Capacity scaling O(E2 log U)
• ?-residual network ??? residual network
  ???????????????????????????????? ?
• ????? ? 2lg U ???????? ? ??????????????????? 1
• Edmonds ??? Karp ????
• ???????????????? (????????????????) O(VE2)
• ?????????????????????????? O(V2 lg U lg V)

(U ????????????????????????????)
23
?????? max-flow

• Augmenting-path method
• Preflow-push method

24
Preflow-Push Method

• augmenting path method ???????????????????????????
  ????? ? ??????????????????????????????????????? ?
  ??????????? (maintain feasibility, aim for
  optimality)
• preflow push method ???????????????????? ? ?????
  sink ???????? ? ??????????????????? source
  ??????????????????????????????????? (start with
  super optimality, aim for feasibility)

25
Original Network
6
5
4
3
2
14
1
s
x
y
z
t
12
5
8
10
0
7
16
26
Residual network step 1 - initialize
6
s
5
-26
4
3
14
12
2
1
x
y
z
t
5
8
10
0
12
14
0
0
7
16
27
Residual network step 2 - relabel
6
s
5
-26
4
3
14
12
2
x
1
12
y
z
t
5
8
10
0
14
0
0
7
16
28
Residual network step 3 - push
6
s
5
-26
4
3
14
12
2
x
1
7
y
z
t
5
8
10
0
19
0
0
7
16
29
Residual network step 4 - push
6
s
5
-26
4
3
14
12
2
x
1
0
y
z
t
5
8
10
0
19
0
7
7
9
7
30
Residual network step 5 - relabel
6
s
5
-26
4
3
14
12
2
x
y
1
0
19
z
t
5
8
10
0
0
7
7
9
7
31
Residual network step 6 - push
6
s
5
-26
4
3
14
12
2
x
y
1
0
11
z
t
5
8
10
0
8
7
7
9
7
32
Residual network step 7 - relabel
6
s
5
-26
4
3
14
12
2
x
y
z
1
0
11
8
t
5
8
10
0
7
7
9
7
33
Residual network step 8 - push
6
s
5
-26
4
3
14
12
2
x
y
z
1
0
11
0
8
t
5
8
2
0
15
7
9
7
34
Residual network step 9 - relabel
6
s
5
-26
4
14
3
y
12
2
11
x
z
5
8
1
0
0
8
t
2
0
15
7
9
7
35
Residual network step 10 - push
6
s
5
-26
4
14
3
y
12
2
6
x
z
5
8
1
5
0
8
t
2
0
15
7
9
7
36
Residual network step 11 - push
6
s
5
-26
4
14
3
y
12
2
6
x
z
5
8
1
0
0
8
t
2
0
20
7
4
12
37
Residual network step 12 - relabel
y
6
6
s
14
5
-26
4
5
8
3
12
2
x
z
1
0
0
8
t
2
0
20
7
4
12
38
Residual network step 13 - push
y
6
6
0
s
8
5
-20
4
5
8
3
12
2
x
z
1
0
0
8
t
2
0
20
7
4
12
39
??????

• e(u) ??? excess ????? u???????????????? u
  ?????????????????????
• active node ?????????? e(u) gt 0(??????? s ??? t,
  e(s) ? 0, e(t) ? 0 ????)
• ?????????????????????
• ??? flow ??????????????????????? (sink
  ??????????)
• ?????? (u,v), d(u) ? d(v) 1
• ????? active node u ???? push ?????? e(u)
  ?????????? "??????" v ?????? d(u) d(v) 1
  (?????????????????? sink)

40
Preflow-Push Method
41
??? distance label ???????

• ??? d(i) 0 ???
• ?? reverse breadth-first search ??? t??? d(t)
  0 ?????? d(i) k ?????? k ?????????????????? i
  ?? breadth-first tree

42
??????? active nodes ???????

• FIFO preflow-push algorithm O(V3)
• Highest label preflow-push algorithm O(V2E1/2)
• Excess scaling algorithm
• O(VE V2logU)

43
FIFO Preflow-Push

• ??? queue ????????????? active nodes
  ??????????????? push flow
• ???????? relabel ?????????????? queue
• ??? u ?????????? push, ??????????? active node
  ???? ???? push ??????????? e(u) 0 ??????????
  relabel

44
FIFO ????????
Pic 1
Pic 2
Pic 4
Pic 3
45
???????? (???)
Pic 6
Pic 5
Pic 8
Pic 7
46
???????? (???)
Pic 9
Pic 10
Pic 11
Pic 12
47
???????? (???)
Pic 13
Pic 14
Pic 15
Pic 16
48
???????? (???)
Pic 17
Pic 18
49
Highest-Label Preflow-Push

• ????? active node ??????????????? push

50
?????????
51
?????????
52
???????????????? sources ??? sinks
53
?????????????????????????
54
???????????????????????????????????????