1 Algorithme naïf 2 Diviser pour régner

E n n
x∈E Ex={y∈E|y=x}
n/2n2
cxExx
E
i n
c←0
j n
E[j] = E[i]
c←c+ 1
c > n/2
E[i]
n2
E
E E1E2n/2n

E E
E
1 2
n/2 n/2
x E x
E1E2x E1E2E
cx≤n/4 + n/4 = n/2
E1E2
E
Majoritaire(i, j) (x, cx)x
E[i..j]cx(−,0)
E[i..j]Majoritaire(1, n)Occurence(x, i, j)
x E[i..j]
Majoritaire(i, j)
i=j
(E[i],1)
(x, cx) = Majoritaire(i, b(i+j)/2c)
(y, cy) = Majoritaire(b(i+j)/2c+ 1, j)
cx6= 0
cx←cx+Occurence(x, b(i+j)/2c+ 1, j)
cy6= 0
cy←cy+Occurence(x, i, b(i+j)/2c)
cx>b(j−i+ 1)/2c
(x, cx)
cy>b(j−i+ 1)/2c
(y, cy)
(−,0)
C(n)
n
C(n) = 2(C(n
2) + n 2)C(1) = 0
C(n) = nlog2(n)
a≥1b > 1f(n)T(n)
T(n) = aT (n/b) + f(n)T(n)
f(n) = O(nlogba−) > 0T(n) = Θ(nlogba)
f(n) = Θ(nlogba)T(n) = Θ(nlogbalog n)
f(n) = Ω(nlogba+) > 0af(n/b)≤cf(n)
c < 1n T (n) = Θ(f(n))

f(n) = n= Θ(n)C(n) = Θ(nlog22log n
E
p > n/2x x
p E x n −p E
x z 6=x
cz≤n−p < n/2x
x
n−1
x p =cx
Candidat −majoritaire(E)
E(x, p)
Candidat −majoritaire(E)
E x
(x, 1)
E E1E2n/2
Candidat −majoritaire(E1) (x, p)
Candidat −majoritaire(E2) (y, q)
E1E2
(y, q)
(y, q +n
4)
(x, p)
(x, p +n
4)
(x, p) (y, q)
x6=y
p > q
(x, p +n
2−q)
p < q
(y, q +n
2−p)
p=q
x y cx≤n
2cy≤n 2
x=y
(x, p +q)

n C(n)
n C(n) = 2C(n
2)+1 C(1) = 0 C(n) = n−1
E
n−1
n−1
2n−2
n
k
≤ dk
2e
M
m M > m
dk
2e
dk
2e
n
C
C
C

C
C
k−1k k ≤n
C≤ dn−1
2e=bn 2c
C
C
C
C
C
C
C
C C
(n−1) −1
C≤ dn
2e −1
C