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Dans le document CAHIER DU LAMSADE 197 (Page 21-30)

Triangle de dimension n

k(h(j)-1)

n

a l = a j

a j

a l

?+







P(a l ) = a k(l)

l = l + 1;

si l ≤ ( j + n − 1) , aller en (2) .

I(a l ) = a h(l)

P (a −1+h(l) ) = a k (−1+ h ( l )) si ∃ i, C i, −1+h(l) ∈ {??,?+}

I(a −1+h(l) ) si ∃ i, C i,−1+h(l) ∈ {??,?−}

P ( a l ) si ∃ i, C i,l ∈ {?? , ?+}

i = l + 1, . . . , − 2 + h(l)

I(a i ) si ∃ i, C i,l ∈ {??,?−}

P ( a i ) si ∃ i, C i,l ∈ {?? , ?+}

l = h(l)

l ≤ (j + n − 1)

Z

Z

W 0

M

W

◦

Z

P(a j )

I(a j )

a i

a j

a l

i < j < l

P (a i ) = a k(i) P ( a l ) = a k(l)

P (a j )

Z

[ a k(i) ,a k(l) ]

◦

W 0

W

Z

a j

a j+1

M

j

j + 1

j

j + 1

a j+1

a j







∀ i ∈ {1, . . . ,j − 2},

C j,i = C j+1,i

P (a j ) = P (a j+1 )

I ( a j ) = I ( a j+1 )

P (a j )

P (a j+1 )

Z

[ a k(j −1) ,a k(j+1) ]

[ a k(j) ,a k(j+2) ]

[a k(j −1) ,a k(j+2) ]

Z

k(j) > k(j + 1)

W 0

a j

a j+1

a j+1 ,a j −1

a j+2 ,a j

a j ,a j−1

a j+2 ,a j+1

I(a j )

I(a j+1 )

a j+1

a j



 

a _l = a _j

a _j

a _l

P(a l ) = a _k(l)

I(a _l ) = a _h(l)

P (a _−1+h(l) ) = a _k ₍₋₁₊ _h ₍ _l ₎₎ si ∃ i, C i, −1+h(l) ∈ {??,?+}

I(a −1+h(l) ) si ∃ i, C _i,−1+h(l) ∈ {??,?−}

P ( a _l ) si ∃ i, C _i,l ∈ {?? , ?+}

I(a i ) si ∃ i, C _i,l ∈ {??,?−}

P ( a _i ) si ∃ i, C _i,l ∈ {?? , ?+}

W ⁰

P(a _j )

I(a _j )

a _i

a _j

a _l

P (a i ) = a _k(i) P ( a _l ) = a _k(l)

[ a _k(i) ,a _k(l) ]

W ⁰

a _j

a _j+1

a _j+1

C _j,i = C _j+1,i

I ( a _j ) = I ( a _j+1 )

P (a _j+1 )

[ a _k(j ₋₁₎ ,a _k(j+1) ]

[ a _k(j) ,a _k(j+2) ]

[a _k(j ₋₁₎ ,a _k(j+2) ]

W ⁰

a _j

a _j+1

a _j+1 ,a _j ₋₁

a _j+2 ,a _j

a _j ,a _j−1

a _j+2 ,a _j+1

I(a _j+1 )

a _j+1

a _j

h(j + 1) = h(j) − 1 a _h(j) Ia _h(j+1)

C _h(j),i = C _h(j+1),i P (a h(j) ) = P (a h(j+1) )

I(a _h(j) ) = I(a _h(j+1) )

k(j + 1) = k(j) − 1 a _k(j) Ia k(j+1)

C _k(j),i = C _k(j+1),i P (a _k(j) ) = P (a _k(j+1) )

[a _k(j ₋₁₎ ,a _k(j+1) ]

[a _k(j) ,a _k(j+2) ]

[a k(j − 1) ,a _1+k(j+1) ]

[a ₋ 1+k(j) ,a _k(j+2) ]

a _k(j) Ia _k(j+1)

W ⁰

a _k(j)

a _k(j+1)

a _k(j) ,a ₋ _2+k(j)

a _1+k(j) ,a _k(j+1)

∀ a _j ∈ A

g (a j ) = x _j

q[x j ] = x _h(j) − x j

p[x j ] = x _k(j) − x _j − ε (avec 0 < ε < 1)

P (a j ) = a _k(j)

a _j

I(a j ) = a _h(j)