Algorithme de Bellman-Ford

G= (S, A, w)s G
s t
t G
S= [|1, n|]s= 1 t S δ(t)
1t δ(t) = ∞t1
k∈δ(t, k) 1 t k
δ(t, n −1) = δ(t)
δ(t, 0) = 0t= 1
∞
k
δ(t, k) = min δ(t, k −1),min
u∈S|(u,t)∈Aδ(u, k −1) + w(u, t)
(G)
T[1 . . . n; 0 . . . n −1] ←n×n∞
T[1,0] ←0
16k6n−1
t∈[|1, n|]
T[t, k]←min T[t, k −1],min
u∈S|(u,t)∈AT[u, k −1] + w(u, t)
T[·, n −1]
O(n2)
O(n2|A|)
u(u, a)a=t

(G)
t6= 1
d[t]← ∞
d[1] ←0
16k6n−1 (1)
(u, t)∈A(2)
d[t]←min {d[t], d[u] + w(u, t)}
d
a1. . . arA(2)
a1a2ar16k6n−1
d0
kd k
i∈[|1, r|]di
kd k
ai
∀06k6n−1,∀t∈S, ∀06i6r, ∀, δ(t)6
(1)
di
k[t]6
(2)
δ(t, k)
(1) di
k[t]
1t k
(2) k
k= 0 δ(t, 0) = ∞t6= 1 ∀i, di
0[1] 6d0 0[1] = 0
k∈∗k−1
∀t∈S, ∀16i6r, di
k−1[t]6δ(t, k −1)
d0
k= min d0 k−1[t],min
i|∃u,ai=(u,t)∈Ad(u,t)
k−1[u] + w(u, t)
6min δ(t, k −1),min
i|∃u,ai=(u,t)∈Aδ(u, k −1) + w(u, t)
=δ(t, k)
∀i∈[|1, r|], di
k[t]6d0 k[t]
O(|S|)

O(|S||A|)
G
1
(u, t)∈A
d[u] + w(u, t)< d[t]
c= (u0. . . uk)u0=uk
1
k
X
i=1
w(ui−1, ui)<0
∀(u, t)∈A, d[u] + w(u, t)>d[t]
∀16i6k, d[ui−1] + w(ui−1, ui)>d[ui]
k
X
i=1
d[ui]6
k
X
i=1
(d[ui−1] + w(ui+1, ui))
=
k
X
i=1
d[ui−1] +
k
X
i=1
w(ui+1, ui)
=
k
X
i=1
d[ui]u0=uk
06
k
X
i=1
w(ui+1, ui)