Arbres binaires de recherche optimaux

K= (k1, . . . , kn)n kipi
kin+ 1 d0, . . . , dn
K d0< k0dn> kn
∀i∈[|1, n −1|]dikiki+1
diqidi
T K ki
di
n
X
i=1
pi+
n
X
i=0
qi= 1
T
[T] =
n
X
i=1
(T(ki) + 1) ·pi) +
n
X
i=0
(T(di) + 1) ·qi
= 1 +
n
X
i=1
T(ki)·pi+
n
X
i=0
T(di)·qi
ki, . . . , kj16j6n
ki, . . . , kjdi−1, . . . , dj
T T 0T0
Ti,j ki. . . kjkr
ki. . . kr−1kr+1 . . . kjr=i
ki. . . ki−1di−1r=j kj+1 . . . kj
dj
0 1
1 2
T0T

i>1i−16j6n e[i, j]
ki, . . . , kj
j=i−1e[i, i −1] = qi−1
j6i
1
w(i, j) =
j
X
l=i
pl+
j
X
l=i−1
ql
e[i, j] = min
i6r6j{pr+ (e[i, r −1] + w(i, r −1)) + (e[r+ 1, j] + w(r+ 1, j))}
w(i, j) = w(i, r −1) + pr+w(r+ 1, j)
e[i, j] = (qi−1j=i−1
min
i6r6j{e[i, r −1] + e[r+ 1, j] + w(i, j)}i6j
e[n+ 1, n]
dn
16i6j6n
racine[i, j]r kr
ki. . . kj
w(i, i −1) = qi−1∀j>i, w(i, j) = w(i, j −1) + pj+qj

p1. . . pnq0. . . qnn
e racine
e[1 . . . n + 1,...n]w[1 . . . n, 0. . . n]racine[1 . . . n, 1. . . n]
16i6n+ 1
e[i, i −1] ←qi−1
w[i, i −1] ←qi−1
16l6n l
16i6n−l+ 1
j=i+l−1
e[i, j]← ∞
w[i, j]←w[i, j −1] + pj+qj
i6r6j
t=e[i, r −1] + e[r+ 1, j] + w[i, j]
t < e[i, j]
e[i, j]←t
racine[i, j]←r
e racine
O(n3)