Travaux dirigés 2: Diviser-pour-régner - LAMSADE

T(n) = aT (n/b) + f(n)
n a n/b bn/bc dn/be
f(n)
f(n) = O(nlogba−ε)ε > 0T(n) = Θ(nlogba)
f(n) = Θ(nlogbalogkn)k∈NT(n) = Θ(nlogbalogk+1 n)
f(n) = Ω(nlogba+ε)ε > 0n af(n/b)≤
cf(n)c < 1T(n) = Θ(f(n))
g(n)T(n) = Θ(g(n)) T(n)
T(n)=8T(n/2) + n3T(n)=8T(n/3) + n2
T(n) = 16T(n/4) + n T (n)=9T(n/3) + n2
T(n)=8T(n/2) + n2T(n)=8T(n/2) + n4log n
T(n) = 17T(n/16) + nlog n T (n)=2T(n/2) + n
A,i,k
i < k
j←i+k
2A,i,j
A,j + 1,k
A,i,j,k
A
9 8 12 3 5 14 6
n
0(n)
A n ≥1
A
T(n)=2T(n/2) + O(1)

F(n)
n= 0
F(n−1) ∗F(n−1)
F n
F
m(n)F(n)
F
G n
G(n)
R←2
i= 1 n
R←R∗R
G
G
?
C=AB A B
2kA,B C
C11 C12
C21 C22!= A11 A12 A21 A22! B11 B12 B21 B22!
Xij X2k−1
i= 1 i= 2 2k−1j= 1
j= 2

C
P1= (A11 +A22)(B11 +B22)
P2= (A21 +A22)(B11)
P3= (A11)(B12 −B22)
P4= (A22)(B21 −B11)
P5= (A11 +A12)(B22)
P6= (A21 −A11)(B11 +B12)
P7= (A12 −A22)(B21 +B22)
C11 =P1+P4−P5+P7
C12 =P3+P5
C21 =P2+P4
C22 =P1−P2+P3+P6
1 3
5 7! 8 4 6 2!
T(n)n
T(n)
1 3
5 7! 8 4
6 2!
O(nlg 7)
n n