Cours de Conception et analyse d`algorithmes I. Algorithme glouton

G= (X, E)v
C G e
e
T e
T−e
(T1, T2)G
T1T2e
e0C e C v(e0)≤v(e)
T−e+e0T
T2
I=E
e I e I
I
V G
GI= (X, I)V
V
I e
GI
GIe e
GI−e2

e I
e={x, y}x e
y
2
O(m2)m
G m log n2
n
s
(s) = w
.
O(n2)
s[i, j]i
j s =s[1, n]
t(i)s[1, i]
i t(0) =
t(i) = ∨
1≤j≤i( (s[j, i]) ∧t(j−1)).
O(n2)
M
O(Mn)2
t(i)
p(i)i t(i)j
(s[j, i]) ∧t(j−1)
p(n)
2

S
a11x1+. . . +a1nxn≤b1
am1x1+. . . +amnxn≤bm
j x
aj1x1+. . . +ajnxn=bjFS
S
x∗S
aj1x∗
1+. . . ajnx∗ m=bjj∈F
S
j /∈F xjaj1xj
1+. . .+ajnxj
n<
bjk=m− |F|x∗=1
kPj /∈Fxj
2
S
x∗
j∈ {1, . . . , m}Sminxaj1x1+
. . . +ajnxnxj
x∗2
A m ×n0 1
x0 1
Ax =1 1 m
•k G
G k
•I={i, i + 1, i + 2}J={j, j + 1, j + 2, k, `}
110000
101010
010101
i x
xk·x`= 0

(G, k)G n
m A 3m+n+ (n−k)
5m+n+n(n−k)
3m i
i= 0, . . . , m −1A
{3i+ 1,3i+ 2,3i+ 3} {5i+ 1,5i+ 2,5i+ 3,5i+ 4,5i+ 5}
3i+ 1
j j = 1, . . . , n 3m+j
(3m+j, 5m+j) (3i+2,5m+j)
i j (3i+ 3,5m+j)i
j
n(n−k)
4m+n+pq 3m+p
3m+n+q
n−k n(n−k)
k
k2
k
k X ={x1, . . . , xn}
d(xi, xj)i, j d(x, y)≤d(x, z) =
d(z, y) (x, y, z)
k G1, . . . , Gk
max
`min
xi,xj∈G`
d(xi, xj)
y1X
i k yiyj
k yix
d(x, yi)
yk+1 {y1, . . . , yk}r= inf1≤i≤k(d(yk+1, yi))
yk+1
2r
k

G= (X, E)f:E→N∗
GfG2f(e) + 1
e2f(e)
G
f Gf
Gf√3
2π/3
G
Gf
M(Gf)Gf
G k
M(Gf)k+Pef(e)
M(Gf)
d0= (1 + √3)/√2d0
k
d0P=NP