9
N: nains de jardin
I: peinture intérieure E: électricité
P: plomberie
T: toit
M: murs
F: fondations
T: fenêtres
X: peinture extérieure
t1 , . . . , tn
ti < tj
i< j O(n log n)
2
i
h
d
b
a
f
g
d
e
d
e
f
c
a
b
c
b
a
p = ∞ P (t)
t
∅ P (∅) = −∞ p P (t) d := 1 p p d
td,k
d d := d + 1
p = 1
p = ∞
k
td,1 = ∅
P (td,1 )
t
>d t ≤ t P (t ) ≤ P (td,1 )
t
>d
P (t) ≤ P (td,1 ) d
b
a
P (t) :=
t
2 P (t) :=
t
3 P (t) t
P (t) := 0
1, 2 ,
P (td,2 ) < P (t)
d td,2 = ∅
t
td,2
td,1 < t P (t ) ≥ P (td,1 )
t
td,2
T
d0 := 0
t
T P (t ) ≥ P (td,1 )
t
d1 , d2 , . . . B0 , B1 , B2 , . . .
Bk := td
k +1,1
, td
k +1,2
, . . . , td
k+1 ,1
B0
. B1
2 2
2 2
p
3
f
e
a
b
c
d
3 2
3
j
k
l
k
j
g
h
i
g
h
i
d
e
f
d
e
f
a
b
c
a
b
c
3
d
O
O
p
D D
2
n
1 3
2
p
O(n + m)
p
∞
O(n + m)
m n I i ti
i
C i
i
C pi
i C
:=
(C i )
i
pi,j
(i, j )
pik
i
k
ri
i
di
i T i :=
T i
i
U i
U i = 1
(0, C i − di ) 0
wi
i tj − ti ≥ aij
aij <
4
a a
2
a
1
(a) a
1 2
i
p1 , . . . , pn [ri , di ]
m A,B,C,D
pA = p B = p D = 2 pC = 4 dA = d D = 5 dB = 3 dC = 4 rA , rB , rC = 0 rD = 2
5
[0, 1] [1, 2] [2, 3] [3, 4] [4, 5]
ri
(di ) −
di
(ri )
p1 , . . . , pn
m
C max m := 3 n := 5 p1 := 10 p2 := 8 p3 := 4 p4 := 14 p5 := 1
B :=
i
B ≤ C max
pi ,
1 m
n
pi
.
i=1
B = C max m := 3 n := 5 p1 := 10 p2 := 8 p3 := 4 p4 := 14 p5 := 1
pi ≤ B
B
B :=
n i=1 pi ≤
i
mB pi ,
t := 0 k := 1
1
m
(
n i=1 pi )
.
i t + pi ≤ B t := t + pi
i
k
t
t + pi
i pi − (B − t)
0
k
t B k := k + 1
m := 4
k + 1 t := p i − (B − t)
( p1 , . . . , p7 ) := (5 , 10, 2, 8, 3, 4, 20)
m−1
p1 , . . . , pn
m
C max
i
pi
C max
i
j
pij
4
pij
3
C max
xij
i
j
xij m
xij ≤ C max,
i = 1, . . . , n
j =1 n
xij ≤ C max ,
j = 1, . . . , m
i=1
m
j =1
xij ≥ 0,
xij = 1, pij
i = 1, . . . , n
i = 1, . . . , n
j = 1, . . . , m
C max
xij C
:= 9 C
xij
C
xij ci ci := C max − lj := C max −
m j =1 xij n i=1 xij
1
C
M lj
i j
M M l
M =
λk P k ,
k=1
P k
k λk = C
k
λk P k
i
j
ti
tj tj − ti ≥ aij
{1, 2, 3, 4, 5} i
pi
2
3 2
3
4
1
5
2 3
0 t6 − t0
I
6
1
2
1
2
−1
3
i
j
C ij
i
C ij l(i, j ) := −∞
C ij
C ij l(i, j )
i
l(i, j )
j i
ri
c ri := l (0, i)
j
ri
0
O(m)
m ti ri O(n3 )
ri
f i := l (0, n + 1) − l(i, n + 1). ri ≤ ti ≤ f i f i − ri
i f i − ri = 0
i i
ti tn+1 − t0
ti
1
2
