Algorithmique - Travaux Dirigés - Corrigés Ecole

n(n−3) (n−2)
P=hv0, ..., vniv0, . . . , vn
P w(i, j, k) = kvivjk+
kvjvkk+kvkvikvivjvk
1≤i < j ≤n t[i, j]
hvi−1, ..., vjit[i, i] = 0 1 ≤i≤n
t
n≥3
n= 3 n−2 = 1 n−3 = 0
n∈N, n ≥3
n n
(i+ 1) (j+ 1) i+j=n, i ≥2, j ≥2
(i−2) j−2
(i−2) + (j−2) + 1 = n−3 +1
(i−1) + (j−1) = n−2
t vi−1vj
vkt[i, j] =

j + 1 cotes
i + 1 cotes
mini≤k≤j−1(t[i, k]+t[k+1, j]+w(i−1, k, j)) t[i, i]
j=i+1 t[i, i+1] = w(i−1, i, i+1)
t[1, n]t[k, j], i+
1≤k≤j t[i, k], i ≤k≤j−1
(i, j)
(1, n)t[i, j]i n
j≥i(j−i)
0
n
n
t[i,j]
1
1
j
i
t[1,n]
(i, j)

i0n
t[i, i]←0
d1n−1
i1n−d
t[i, i +d]←mini≤k≤j−1(t[i, k] + t[k+ 1, j] + w(i−1, k, j))
t[1, n]
t[i, j] 2(j−i)j−i−1
min
An=Pn−1
d=1 ((n−d)·(2d)) (n−d)
t[i, j]j−i=d2d
t[i, j]j−i=d
An= 2n·
n−1
X
d=1
d−2
n−1
X
d=1
d2
= 2n·n·(n−1)
2−2(n−1) ·n·(2n−1) 6
=(n−1) ·n·(n+ 1)
3
∼n3/3 = Θ(n3)
Tn
Tn=An/2−Cn
Cnt[i, j]
Tn=An/2−
n−1
X
d=1
(n−p)
= Θ(n3)
Θ(n3)
w
C3
n+1 = Θ(n3)
Θ(n3)
(v0vi)2≤i≤n−1
n−3 Θ(n3)

n
i(xi, yi, zi)
xiyizi
xi, yi, zi
xi≤yi≤zi
xj, yj, zjxj< yiyj< zi
xj≤yj≤zjxiyizi
xi
2n
Li, li, hiLi≥li
3n n
xi, yi, zixi< yi< zi
zi, yi, xizi, xi, yiyi, xi, zi
Li, li, hiLj, lj, hjLi< Lj
li< lj
∀i∈ {1, . . . , 3n}, Hi=i

∀i∈ {1, . . . , 3n}, Hi=hi+Max1≤j≤3n(Hj/Lj> Lilj> li)
3n Lili
O(nlog(n))
∀i∈ {1, . . . , 3n}, Hi=hi+Max1≤j<i(Hj/Lj> Lilj> li)
Hii3n H1=h1
Hi3i−3
Hi, i ∈1, . . . , 3n
H=Max1≤i≤3n(Hi)
3n−1O(n2)
O(n)
←
i1n
xiyiziliste
liste = (Li, li, hi)i∈{1,...,3n}Li, li
H1←h1
i2 3n
Hi←hi+Max1≤j<i(Hj/Lj> Lilj> li)
H←Max1≤i≤3n(Hi)
n l1l2ln
M
i j i ≤j
M−j+i−Pj
k=ilk