Travaux dirigés 1 : Notations asymptotiques, Complexité

Ω(g(n)) = {f(n) : 0 ≤cg(n)≤f(n)∃c > 0∃n0∈N∀n≥n0}
Θ(g(n)) = {f(n) : 0 ≤c1g(n)≤f(n)≤c2g(n)∃c1,c2>0∃n0∈N∀n≥n0}
O(g(n)) = {f(n) : 0 ≤f(n)≤cg(n)∃c > 0∃n0∈N∀n≥n0}
f(n) = O(g(n)) f(n)∈O(g(n)) Ω Θ
O(1)
n
n f(n) =
O(g(n)) O(g(n))
O(g(n)) Ω(g(n))
f(n) = Ω(g(n)) Θ(g(n))
O(g(n)) Ω(g(n))
Θ(f(n)) = Ω(f(n)) ∩O(f(n)) f g
f(n) = O(g(n)) Ω(n3) Θ(n3)
n m
A(n,m)
i←1j←1
(i≤m) (j≤n)
i←i+ 1
j←j+ 1
B(n,m)
i←1j←1
(i≤m) (j≤n)
i←i+ 1
j←j+ 1
C(n,m)
i←1j←1
(j≤n)
(i≤m)
i←i+ 1
j←j+ 1
D(n,m)
i←1j←1
(j≤n)
(i≤m)
i←i+ 1
{j←j+ 1; i←1}
Θ(·)n≤m
n>m

n2=O(10−5n3) 25n4−19n3+ 13n2=O(n4) 2n+100 =O(2n)
O(n n)O(2n)
O(n)O(1) O(n2)O(n3)O(n)
f g S T N→N
f(n) = O(g(n)) g(n) = Ω(f(n))
f(n) = O(g(n)) f(n) + g(n) = O(g(n))
f(n) + g(n) = Θ(max{f(n),g(n)})
O(f(n) + g(n)) = O(max{f(n),g(n)})
S(n) = O(f(n)) T(n) = O(g(n))
f(n) = O(g(n)) S(n) + T(n) = O(g(n))
S(n)T(n) = O(f(n)g(n))
A
j←2longueur[A]
cl´e←A[j]
BA[j]A[1, . . . ,j −1]
i←j−1
i > 0A[i]> cl´e
A[i+ 1] ←A[i]
i←i−1
A[i+ 1] ←cl´e
?
j A[1..j]
k k logkx= (ln k)−1ln x
O(logkn)
O(logk+1 n)
n t(n) = dlog2ne
a:= t(n)b:= t(m)
a b n m