Path Finding Algorithm
Bellman-Ford Algorithm
回顧
Bellman-Ford Algorithm
- 對圖進行 V-1 次鬆弛 (relax)
假設圖上有 V 個頂點
（因最多經過 V-1 條邊，必定到達終點）
- 假設進行 V 次仍能降低 cost，則表示有負權環 (negative cycle)
（通常還未進行到 V-1 次時，就已不會降低 cost）
- 常見優化：假設進行 M 次 (M < V-1) 即不降低 cost，則可直接終止演算法
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
V=7
|V-1|=6
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 6 | ∞ | ∞ | ∞ | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 6 | 5 | ∞ | ∞ | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 6 | 5 | 5 | ∞ | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 6 | 5 | 5 | 5 | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 5 | 5 | 5 | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
5+(-2)<6
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 5 | 5 | 5 | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
5+1>5
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | ∞ | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
5+(-2)<5
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | ∞ |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 8 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
4+3<8
First round completed
1st
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
Second round start
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
0+6>3
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
0+5>3
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
0+5=5
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 3 | 3 | 5 | 2 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
3+(-1)<5
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
3+(-2)<3
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
3+1>2
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
5+(-2)=3
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 7 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
5+(-1)=4
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 5 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
2+3<7
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 5 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
4+3>5
Second round completed
2nd
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 5 |
| 0 | 1 | 3 | 5 | 0 | 4 | 3 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
2nd
3rd
| a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|
| 0 | 3 | 3 | 5 | 5 | 4 | 7 |
| 0 | 1 | 3 | 5 | 2 | 4 | 5 |
| 0 | 1 | 3 | 5 | 0 | 4 | 3 |
| 0 | 1 | 3 | 5 | 0 | 4 | 3 |
a
b
c
d
e
f
g
6
5
5
-2
-2
-1
1
-1
3
3
(a,b)(a,c)(a,d)(b,e)(c,b)(c,e)(d,c)(d,f)(e,g)(f,g)
EdgeList:
1st
2nd
3rd
4th
Nothing changed
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| ∞ | ∞ | ∞ | ∞ |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | ∞ |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | ∞ |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | 5 |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
1st
2nd
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
| 0 | -2 | 8 | 5 |
(c,b)(d,c)(a,d)(a,b)
EdgeList:
1st
2nd
3rd
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | ∞ |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | ∞ |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | ∞ |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | ∞ | ∞ | 5 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
4+5>5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
1st
2nd
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
| 0 | -2 | 8 | 5 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
1st
2nd
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
| 0 | -2 | 8 | 3 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
1st
2nd
-2+5<5
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
| 0 | -2 | 8 | 3 |
| 0 | -2 | 8 | 3 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
1st
2nd
3rd
a
b
c
d
4
5
3
-10
| a | b | c | d |
|---|---|---|---|
| 0 | 4 | ∞ | 5 |
| 0 | 4 | 8 | 5 |
| 0 | -2 | 8 | 3 |
| 0 | -2 | 6 | 3 |
(c,b)(d,c)(a,d)(a,b)(b,d)
EdgeList:
5
1st
2nd
3rd
3+3<8
Negative cycle detected
b
c
d
3
-10
5
Negative cycle
5+3-10=-2
References
Bellman Ford Algorithm
By Travor Lee
Bellman Ford Algorithm
- 185
