Cours - LIFL

(S, A, P ) (S, A)
P
c=v0v1, v1v2, ....vk−1vk
P(c) = Pk−1
i=0 P(vivi+1) 1
s b
a b δ(s, b)a b
δ(s, b)
(s, b)
s b δ(s, b)
b s
δ(s, b)
(s, b)
s b δ(s, b) = +∞
δ(s, b) = −∞
G= (S, A, P )s
u, v δ(s, v)≤δ(s, u) + p(u, v)
δ(s, v) = 0 s=v
δ(s, v) = min{δ(s, u) + p(u, v)/(u, v)∈A}s v
s v |S| − 1
∞;
δ(s, v)

δ(s, v);
O(|S|∗|A|)O(|A|)
|S| − 1
D(s, noeud)
δ(s, v)
δ(s, b)
O(|A|+|S|)
δ(s, n)

∞;
δ(s, b)
b δ(source, b)b
δ(source, b)D(b)δ(source, b)n
b D(b) = δ(source, b)
b D(b)source b
O(1) O(|S|)O(|S|2+|A|)O(|S|2)
O(log(|S|)) O(|S|log(|S|) +|A|log(|S|)O(|A|log(|S|))
O(|A|+|S|log(|S|))
O(1)
n D(n)
1|S|
p p(i, j) +∞(i, j)i=j(i, j)

dl(i, j)i j l
d0(i, i) = 0, d0(i, j) = +∞i6=j
dl(i, j) = min(dl−1(i, j), min(k,j)∈A(dl−1(i, k) + p(k, j)))
δ(i, j) = d|S|−1(i, j)
|S| − 1
dl(i, j)DlDl
Dl−1O(|S|3)O(|S|4
min Dl
PlP
P|S|−1O(log(|S|)O(|S|3log(|S|)
1 1 2
ak(i, j)i j k
i j
a0(i, j) = p(i, j)
ak(i, j) = min(ak−1(i, j), ak−1(i, k) + ak−1(k, j))
k
a|S|(i, j)a|S|(i, j) = δ(i, j)
ak(i, k) = ak−1(i, k)ak(k, j) = ak−1(k, j)
δ(i, j)
O(|S|3)