Cours de conception et analyse d`algorithmes

m n
p1, . . . , pn
α α = 2 α= 2 −1/m
G= (V, E)n=|V|
G
G= (V, E)n=|V|k
T G
k T
F⊂E G = (V, E)δF(x)
x F F x ∆(F) = maxx∈VδF(x)
F∆∗(G) ∆(T)T
G
∆∗(G)≤k

k≥∆(E)
(G, k)
k k
G(G, k)
k= 2
T G (u, v)G\T
w T [u, v]u v T δT(w)w T
max(δT(u), δT(v)) + 1
G T G
T0G
T
∆∗(G)
∆∗(G)
k T G T1, . . . , Tk+1
T S
G G TiTj
S
T0G k/|S|∆∗(G)≥k/|S|
T
S
SiG i T i
i(i−1)|Si|+ 1 T
SiT
G
Si−1
c > 1i≥∆∗(T)−logcn
0<|Si−1| ≤ c|Si|
T c∆∗(G) + dlogcne
(G)G
S
G S
x0G
x0
x0
x0

x x y
{x, y}y
n m
G
k G k
G k
m{s1, . . . , sm}
T= (t1, . . . , tn)ti∈T sj
c(i, j)
T C∗(T)
(α)
α > 0β≥1α > 0
T= (t1, . . . , tn)
α < C∗(T) (α)k
k≤n k −1
αβ
α≥C∗(T) (α)n
αβ
(α)
4β
(α)
α¯c(i, j) = c(i, j)/α
Si={sj∈S|¯c(i, j)<1}
α
tiSi
T(P)n·m

X
ti∈T
X
sj∈Si
yi,j
∀ti∈T, X
sj∈Si
yi,j ≤1
∀sj∈S, X
ti∈T,sj∈Si
¯c(i, j)yi,j ≤1
xjj= 1, . . . , m zii= 1, . . . , n
α≥C∗(T) (P)
n
n
∀j xj:= 1/(2m)∀i zi:= 1 ∀i, j yi,j := 0
ti
Si=∅xjxj>1
s`∈Si¯c(i, `)x`
tis`
zi:= 1 −¯c(i, j)x`yi,` := 1
x`:= x`(1 + 1
2¯c(i, `))
O(log m)
xj≤3/2
xjsj
n
O(log m)