1 L`algorithme de Bellman-Ford

G= (V, E)w s
s
π[v]v
δ(u, v)u v ∞d[v]
s v
p=hv0, v1, . . . , vkiw(p) = Pk
i=1 w(vi−1, vi)
(G, w, s)
v∈V
d[v]← ∞ π[v]←
d[s]←0
i1|V| − 1
(u, v)∈E(u, v)
d[v]> d[u] + w(u, v)
d[v]←d[u] + w(u, v)π[v]←u
(u, v)∈E
d[v]> d[u] + w(u, v)
z(y, v) 4
z
u
x
v
y
6
7
8
9
7
-3
-4
2
5
-2
(u, v) (u, x) (u, y) (v, u) (x, v) (x, y) (y, v) (y, z) (z, u) (z, x)
w(y, v) = 4
u v

z
u
x
v
y
6
7
8
9
7
-3
-4
2
5
-2
∞
0
∞
∞∞
z
u
x
v
y
6
7
8
9
7
-3
-4
2
5
-2
6
0
∞
∞7
z
u
x
v
y
6
7
8
9
7
-3
-4
2
5
-2
6
0
4
27
z
u
x
v
y
6
7
8
9
7
-3
-4
2
5
-2
2
0
4
27
z
u
x
v
y
6
7
8
9
7
-3
-4
2
5
-2
2
0
4
-27
z
u
x
v
y
6
7
8
9
4
-3
-4
2
5
-2
∞
0
∞
∞∞
z
u
x
v
y
6
7
8
9
4
-3
-4
2
5
-2
6
0
∞
∞7
z
u
x
v
y
6
7
8
9
4
-3
-4
2
5
-2
6
0
4
27
z
u
x
v
y
6
7
8
9
4
-3
-4
2
5
-2
2
0
4
27
z
u
x
v
y
6
7
8
9
4
-3
-4
2
5
-2
2
0
2
-27

O(V)O(V E)
O(E)O(V E)
(u, v)d[v]≤d[u] + w(u, v)
d[v]> d[u] + w(u, v)
d[v]←d[u] + w(u, v)π[v]←u
d[v]≤d[u] + w(u, v)
d[v]> d[u] + w(u, v)d[v]←d[u] + w(u, v)d[v] =
d[u] + w(u, v)
d[v]≥δ(s, v)v
d[v]δ(s, v)
d[v]≥δ(s, v)v∈V
d[v]≥δ(s, v)
d[v] = ∞,∀v∈V− {s}d[s]=0≥δ(s, s)δ(s, s) = −∞ s
0
(u, v)d[x]≥
δ(s, x)x∈V d d[v]
d[v]d[u] + w(u, v)
≥δ(s, u) + w(u, v)
≥δ(s, v)∀(u, v)δ(s, v)≤δ(s, u) + w(u, v)
d[v] = δ(s, v)
d[v]
d
(s, . . . , u, v)
v d[u] = δ(s, u)
(u, v)d[v] = δ(s, v)
d[u] = δ(s, u)
(u, v)d[v]≥δ(s, v)
(u, v)

d[v]≤d[u] + w(u, v)
δ(s, u) + w(u, v)
δ(s, v)
G
s d[v] = δ(s, v)v
s
p=hv0, v1, . . . , vkiG s =v0vk
p(v0, v1) (v1, v2) (vk−1, vk)d[vk] = δ(s, vk)
p
i p
d[vi] = δ(s, vi)i= 0 d[v0] = d[s] =
0 = δ(s, s)d[vi−1] = δ(s, vi−1)
d[vi] = δ(s, vi)
G s
d[v] = δ(s, v)v s
v s s v p =hv0, v1, . . . , vki
v0=s vk=v p |V| − 1k≤ |V| − 1
|V| − 1|E|
i i = 1,2, . . . , k (vi−1, vi)
d[v] = d[vk] = δ(s, vk) = δ(s, v)
v s v
d[v]<∞
s v
d[v] = δ(s, v) = ∞
∞=δ(s, v)≤d[v]
d[v] = ∞=δ(s, v)
G
s d[v] = δ(s, v)v∈V
Gπv π(v)6= (π(v), v)
s G
s
G
s v s d[v] = δ(s, v)
v s
d[v] = ∞=δ(s, v)
d[v] = δ(s, v)v∈V
s

Gπs δ(s, v)v
s s
d d[v]v∈V− {s}
π[v]6=Gπs
Gπs Gπ
Gπ
Gπ
Gπc=hv0, v1, . . . , vkiv0=vkπ[vi] = vi−1
i= 1,2, . . . , k
(vk−1, vk)
c s
(vk−1, vk)π[vi] = vi−1i= 1,2, . . . , k −1
d[vi]d[vi]←d[vi−1]+w(vi, vi−1)d[vi−1]
d[vi]≥d[vi−1]+ w(vi, vi−1)
π[vk]d[vk]> d[vk−1] + w(vk, vk−1)
c
k
X
i=1
d[vi]>
k
X
i=1
(d[vi−1] + w(vi−1, vi))
Pk
i=1 d[vi] = Pk i=1 d[vi−1] 0 >Pk i=1 w(vi−1, vi)
G
Gπ
Gπs
v∈Sπs v Gπ
Gπ∀v∈Vπp s v
Gπs v G p =hv0, v1, . . . , vkiv0=s
vk=v i = 1,2, . . . , k d[vi] = δ(s, vi)d[vi]≥d[vi−1]+w(vi−1, vi)
w(vi−1, vi)≤δ(s, vi)−δ(s, vi−1)
p
w(p)Pk
i=1 w(vi−1, vi)
≤Pk
i=1(δ(s, vi)−δ(s, vi−1))
δ(s, vk)−δ(s, v0)
δ(s, vk)
w(p)≤δ(s, vk)δ(s, vk)
s vkw(p) = δ(s, vk)p
s v =vk
d[v] = δ(s, v)Gπ
G
∀(u, v)∈E
d[p]δ(s, v)
≤δ(s, u) + w(u, v)
d[u] + w(u, v)