Devoir 1

OΩ Θ
OΘ
OΘfi
N c N c1c2OΘ
fi
f1(n)=lg
3(9n3×3n+1)+6lg 3n∈O(?)
f2(n)= n
2n+1 +4n3+ 1000n∈O(?)
f3(n)=4
n+2 +8n2+6n∈Θ(?)
•lg nk=k×lg n
•lg (n1×n2)=lg n1+lg n2
•ai+j=ai×aj
•n≥1a≥2
lg n≤n
n≤an
•x−y=1
xy
•logbn=logab×logan
≤ ≥
lg (n2+1)
n≥2
lg (n2+1) ≤lg(n2+n2)
=lg(2n2)
=lg2+lgn2
=lg2+2lgn
≤lg n+2 lg n
=3lgn
lg (n2+1)∈O(lg n)N=2 c=3
O, Ω,Θ
O
O f(n)∈O(n2) [f(n)+f(n)] ∈O(n2)

←
←←←
←
←
←←
←
←
Θ
O
n
n
X
i=1
i=n(n+1)
2
n
X
i=1
i2=n(n+ 1)(2n+1)
6

• ⇒
• ⇒
• ⇒
• ⇒
• ⇒
• ⇒
Θ

T(1) ∈Θ(1) n
b
T(n)=10T(n/3) + n2
T(n)≤5T(n/2) + n3
T(n)=4T(n/5) + 2
√n+1
T(n)≥9T(n/3) + n2
T(n)=8T(n/7) + n2lg n
lgkn= (lg n)k
k
√n=n
1
k
logxy=logzy
logzx