Growth of Functions
1
Asymptotic Notation
• 충분히 큰 n 에 대하여, 함수 f(n) 을 근사적으로 표현하는 방법
– 알고리즘의 running time 을 표현하기 위하여 사용
• 종류
–
–
–
–
–
Θ : asymptotic tight bound
O : asymptotic lower bound
 : asymptotic upper bound
o : asymptotic strict lower bound
ω : asymptotic strict upper bound
2
Asymptotic Notation
• Θ notation
– Definition
f (n)  ( g (n))
↔ there exist positive constants c1 , c2 , no such that
0  c1 g (n)  f (n)  c2 g (n)
n  no
- Asymptotically tight bound
(Ex)
3
Asymptotic Notation
• O notation
– Definition
f (n)  O( g (n))
↔ there exist positive constants c, no such that
0  f (n)  cg (n)
n  no
- Asymptotic upper bound
Worst-case 의 running time 표현에 적합
-
( g (n))  O( g (n))
(Ex)
4
Asymptotic Notation
• Ω notation
– Definition
f (n)  ( g (n))
↔ there exist positive constants c, no such that
0  cg (n)  f (n)
n  no
- Asymptotical lower bound
Best-case 의 running time 표현에 적합
(Ex)
(Ex) Insertion Sort 알고리즘의 running time:
(n) ~ O(n 2 )
5
Asymptotic Notation
• o notation
– Definition
f n  og n
↔ there exist positive constants c, no such that
0  f ( n )  cg ( n )
- Asymptotic strict upper bound
Upper bound that is not asymptotically tight
lim
n
f (n)
0
g( n )
(Ex)
6
Asymptotic Notation
• ω notation
– Definition
f n  g n
↔ there exist positive constants c, no such that
0  cg ( n )  f ( n )
- Asymptotic strict lower bound
Lower bound that is not asymptotically tight
lim
n
f (n)

g( n )
(Ex)
7
Comparisons of Functions
• Related Properties:
– Transitivity:
f (n) = (g(n)) and g(n) = (h(n))⇒ f (n) = (h(n)).
Same for O, , o, and ω.
– Reflexivity:
f (n) = ( f (n)).
Same for O and .
– Symmetry:
f (n) = (g(n)) if and only if g(n) = ( f (n)).
– Transpose symmetry:
f (n) = O(g(n)) if and only if g(n) = ( f (n)).
f (n) = ω(g(n)) if and only if g(n) = ω( f (n)).
8
Standard notations and common functions
• Monotonicity:
– f (n) is monotonically increasing if m ≤ n ⇒ f (m) ≤ f (n).
– f (n) is monotonically decreasing if m  n ⇒ f (m) ≥ f (n).
– f (n) is strictly increasing if m < n ⇒ f (m) < f (n).
– f (n) is strictly decreasing if m  n ⇒ f (m) > f (n).
• Floor and Ceilings:
• Modular arithmetic:
x – 1 <  x   x   x  < x+1
a mod n = a - a/n n
9
Polynomials, Exponentials and Logarithms
• Polynomials
-
-
a d n d  a d 1 n d 1    a1 n  a o
 
f n   O n k for some k

f n  is polynomial ly bounded
• Exponentials
n
-
-
-
a
(a  1)
exponential 함수는 polynomial 함수에 비하여 급격히 증가
nb
n
b
a  n
 lim n  0  n b  o(a n )
n   a
exponentially bounded => 무한대의 running time 을 갖는 알고리즘
• Logarithms
-
lg n  log 2 n , ln n  log e n
polynomial 함수는 logarithm 함수에 비하여 급격히 증가
lg b n
n  lg n  lim
 0  lg b n  o(n a )
a
n   n
a
-
 
b
f n   O lg k n for some k

f n  is poly - logarithmi cally bounded
10
Polynomials, Exponentials and Logarithms
11