Algorithmes Optimaux et Sous-optimaux de Singleton Consistance d

1 2
1
2
k

P= (X, D, C)
n X ={i, j, . . .}
D={D(i), D(j), . . .}
D(i)d
i e
C={c1, . . . , ce}ck
var(ck)
sol(ck)
var(c) = (i, j)
sol(c)cij
a i
(i, a)ck
(i, a)ck
var(ck)
a i
ck(i, a)
P= (X, D, C)
D
D
AC(P)
P
AC(P)P
P= (X, D, C)
∀i∈X, ∀a∈D(i)
P|i=a= (X, D|i=a, C)D(i)
{a}P|i=a
(i, a)
O(en2d4)
O(end3)
(i, a)
P|i=a
D(j)b
((i, a),(j, b)) (i, a)
((i, a),(j, b))
(i, a)
(i, a)j
b∈D(j) ((i, a),(j, b))
(i, a)
b D(j)
((i, a),(j, b))
(k, c)
k
(i, a) (j, b)
k j
(i, a)
(k, c)
((i, a),(j, b)) D(k)
(i, a)
Σb∈D(j)Σckj ∈CΣc∈D(k)1 = ed2
(i, a)
(j, b) ((i, a),(j, b))
(i, a)
((k, c),(j, b))
(j, b)
O(Σ(i,a)∈Ded2) = O(end3)2

P|i=a
(i, a)
AC(P|i=a)
(j, b)
(i, a)
(j, b)AC(P|i=a)
P|i=a
O(end3)
nd
P|j=b
(i, a)
n2d2
nd
P|i=a
P
nd
P
Pia
Dia
P|i=apropagAC(P, Q)
Q P
P
Pia
P
(P)← ∅
(i, a)∈D
Pia ←P
Dia(i)← {a}Qia ←D(i)\ {a}
¬propagAC(Pia, Qia)
D←D\ {(i, a)}
propagAC(P, {(i, a)})
Pjb (i, a)∈Djb
Qjb ←Qjb ∪ {(i, a)}
← ∪ {(j, b)}
6=∅
(i, a)
propagAC(Pia, Qia )Qia ← ∅
D←D\ {(i, a)}
D(i) = ∅
(j, b)∈D(i, a)∈Djb
Djb(i)←Djb(i)\ {a}
Qjb ←Qjb ∪ {(i, a)}
← ∪ {(j, b)}
a D(i)
P
a i
P|i=aPia
(i, a)
P P
Pkc (k, c)
Pjb
(k, c)
P
Pjb a∈Djb(i) (i, a)
Qjb Pjb
(i, a)
Qia
Pia
(i, a)
D

Pjb (i, a)
Qjb
D
O(end3)
O(end2)
a
D(i) (i, a)
Pia Pia
(i, a)
Pia
Pia
(i, a)
Qia
Pia
(i, a)P
P|i=a(i, a)
nd
O(end2)
nd ·ed nd ·ed2while
nd nd
nd
ed2nd ·ed2
nd nd ·ed2
nd nd
Q
n2d2n < e
O(end3)2
Qia
O(n2d2)
O(end4)
O(end2)
nd P |i=a
O(ed2)
O(ed)
nd·ed
P|i=a

(i, a)Dia
P|i=aQia
Qia
j
(j, b)
Dia
Dia (i, a)
(j, b) (i, a)
(j, b)Dia
Dia
P
Pia
P
Last Last(i, a, j)
D(j) (i, a)cij
P
Pia
P
(i, a)
Pia Dia nil
Pia
Dia Pia
(i, a)
D
P
propagMainAC
(i, a)P
updateSubProblems
P
Qjb
P
(P)← ∅
(i, a)∈D
Dia ←nil Qia ← {i}
← ∪ {(i, a)}
6=∅
(i, a)
a∈D(i)
Dia =nil
Dia ←(D\D(i)) ∪ {(i, a)}
propagSubAC(Dia, Qia)Qia ← ∅
D(i)←D(i)\ {a} ← {(i, a)}
propagMainAC(D, {i},)
updateSubProblems( )
Propag Main/Sub AC D
Q
Q6=∅
j Q
i∈X∃cij ∈C
a∈D(i)
Last(i, a, j)6∈ D(j)
∃b∈D(j), b > Last(i, a, j)∧cij (a, b)
Last(i, a, j)←b
D(i)←D(i)\ {a}Q←Q∪ {i}
← ∪ {(i, a)}
D(i) = ∅
··· propagMainAC propagSubAC
updateSubProblems
(j, b)∈D|Djb ∩ 6=∅
Qjb ←Qjb ∪ {i∈X|Djb (i)∩ 6=∅}
Djb ←Djb \
← ∪ {(j, b)}
propagMainAC
PpropagSubAC
Pia
propagMainAC propagSubAC
propagSubAC