1. nalisis de lgoritmos A A
Informacion tomada del Depto. De sistemas de la Universidad Nacional sede Bogota.

2. OUTLINE
Growth of Functions
Asymptotic notation
Common functions

3. ASYMPTOTIC NOTATION

4. Asymptotically No Negative Functions
f(n) is asymptotically no negative if there exist n0  N such that for every n  n0,
0  f(n) f n0

5. Theta 
 (g(n)) ={ f : N R* | ( c1, c2  R +) ( n0 N) ( n  n0) (0 c1 g(n)  f(n)  c2 g(n)) }
c1  lim f(n)  c2
n  g(n) g
c2g f
c1g n0

6. “f(n) =  (g(n))”  “f(n)   (g(n))”
f is asymptotically tight bound for g or
f is of the exact order of g
Every member of (g(n)) is asymptotically no negative
The function g(n) must itself asymptotically no negative, or else (g(n)) = .

7. Example
Lets show that
We have to find c1,c2 and n0 such that

8. For all n  n0, Dividing by n2 yields
We have that 3/n is a decreasing sequence
3, 3/2, 1, 3/4, 3/5, 3/6, 3/7…
and then 1/2 - 3/n is increasing sequence
-5/2, -1, -1/2, -1/4, -1/10, 0, 1/14
that is upper bounded by 1/2.

9. The right hand inequality can be made to hold for n  1 by choosing c2  1/2. Likewise the left hand inequality can be made to hold for n  7 by choosing c1  1/14.

10. Thus by choosing c1 = 1/14, c2 = 1/2 and n0 =7 then we can verify that

11. O(g(n)) = {f: N R* | ( cR +) (n0N) (n n0) ( 0  f(n)  cg(n) )}
Big O
O(g(n)) = {f: N R* | ( cR +) (n0N) (n n0) ( 0  f(n)  cg(n) )}
lim f(n)  c
n  g(n) cg g f n0

12. “f(n) = O (g(n))”  “f(n)  O (g(n))”
f is asymptotically upper bounded for g or
g is an asymptotic upper bound for f
O (g(n)) is pronounced “big oh of g(n)”

13. We have to find c and n0 such that
Example
Lets show that
We have to find c and n0 such that

14. For all n  n0, Dividing by n yields
The inequality can be made to hold for n  1 by choosing c  a+b. Thus by choosing c = a+b and n0 =1 then we can verify that

15. a+b a b
n0=1

16. Big Omega 
 (g(n)) ={ f: N R* | ( cR +) (n0N) ( n n0) ( 0 cg(n)  f(n) ) }
c  lim f(n)
n  g(n) g cg f n0

17. “f(n) =  (g(n))”  “f(n)   (g(n))”
f is asymptotically lower bounded for g or
g is an asymptotic lower bound for f
O (g(n)) is pronounced “big omega of g(n)”

18. “o(g(n)) functions that grow slower than g ”
Little o
o (g(n)) = { f : N R* | (c R +) (n0N) ( n n0) ( 0  f(n) < cg (n) ) }
lim f(n) = 0
n  g(n)
“o(g(n)) functions that grow slower than g ” g f

19. “f(n) = o(g(n))”  “f(n)  o (g(n))”
f is asymptotically smaller than g
o(g(n)) is pronounced “little-oh of g(n)”

20. Examples
Lets show that
We only have to show n2 n

21. Lets show that
We have 3n n

22. “ (g(n)) functions that grow faster than g ”
Little Omega 
 (g(n)) = { f : N R* | (cR +) (n0N) ( n n0) ( 0  c g(n) < f(n) ) }
lim f(n) = 
n  g(n)
“ (g(n)) functions that grow faster than g ” f g

23. “f(n) =  (g(n))”  “f(n)   (g(n))”
f is asymptotically larger than g
o(g(n)) is pronounced “little-omega of g(n)”

24. Analogy with the comparison of two real numbers
Asymptotic Real
Notation numbers
f(n) O(g(n)) f  g
f(n) (g(n)) f  g
f(n) (g(n)) f = g
f(n) o(g(n)) f < g
f(n) (g(n)) f > g
Trichotomy does not hold

25. Example
f(n)=n
g(n)=n(1+sin n)
(1+sin(n))  [0,2]
Not all functions are asymptotically comparable

26. The running time of an algorithm is (f(n)) iff
Properties
 (f(n))=O(f(n))  (f(n))
The running time of an algorithm is (f(n))
iff
Its worst-case running time is O(f(n)) and
Its best-case running time is  (f(n))

27. f(n)  (g(n)) and g(n)  (h(n)) then f(n)  (h(n)) ;
Transitivity of O,  , 
f(n)  (g(n)) and g(n)  (h(n)) then f(n)  (h(n)) ;
for  = O,  , 
Reflexivity of O,  , 
f(n)  (f(n)) ; for  = O,  , 
Symmetry of 
f(n)  (g(n))  g(n)  (f(n))
Anti-symmetry of O, 
f(n)   (g(n))
f(n)  (g(n))  g(n)  (f(n)) ; for  = O, 
Transpose Symmetry
f(n) O(g(n))  g(n)  (f(n))
f(n) o(g(n))  g(n)  (f(n))

28. f  g  f(n) O(g(n)) order relation
reflexive
anti-symmetric
transitive
f  g  f(n)  (g(n)) order relation
f = g  f(n)  (g(n)) equivalence relation
symmetric

29. Relation between o and O
f(n) o(g(n))  f(n) O(g(n))
Relation between  and 
g(n)  (f(n)) g(n)  (f(n))
o(f(n))   (f(n)) = 

30. Examples
A B
5n n 3n2 + 2
log3(n2) log2(n3)
nlg4 3lg n
lg2n n1/2

31. Examples A B 5n2 + 100n 3n2 + 2 A  (B) log3(n2) log2(n3) A  (B)
nlg4 3lg n A  w(B)
lg2n n1/2 A  o (B)

32. Examples A B 5n2 + 100n 3n2 + 2 A  (B) log3(n2) log2(n3) A  (B)
A  (n2), n2  (B)  A  (B)
log3(n2) log2(n3) A  (B)
logba = logca / logcb; A = 2lgn / lg3, B = 3lgn, A/B =2/(3lg3)
nlg4 3lg n A  w(B)
alog b = blog a; B =3lg n=nlg 3; A/B =nlg(4/3)   as n
lg2n n1/2 A  o (B)
lim ( loga n / nb ) = 0, here a = 2 and b = 1/2  A  o (B)
n

33. Asymptotic notation two variables
O(g(m, n))={ f: N N  R* | ( c R +)( m0 , n0  N)
( n n0) ( m m0)
(f(m, n)  c g(m, n)) }

34. COMMON FUNCTIONS

35. Monotonicity f is monotonically increasing if m n  f(m) f(n)
f is monotonically decreasing if
m n  f(m) f(n)
f is strictly increasing if
m < n  f(m) < f(n)
f is strictly decreasing if
m < n  f(m) > f(n)

36. Floors and Ceilings x floor of x x ceiling of x
the greatest integer less than or equal to x
x ceiling of x
the smallest integer greater than or equal to x
x  R , x-1 < x  x  x < x +1
n  N , n = n = n and n/2 + n/2 = n -1 1
x -2 2
x x

37. x and x are monotonically increasing
x  R and integers a,b > 0
  x/a  /b = x /ab .
 x/a /b =  x /ab .
a /b  ( a + (b-1)) /b.
a /b  ( a - (b-1)) /b.
x and x are monotonically increasing

38. Modular arithmetic
For every integer a and any possible positive integer n,
a mod n
is the remainder of ( or residue) of the quotient a/n
a mod n = a - a /n n.

39. congruency or equivalence mod n
If (a mod n) = (b mod n) we write
a  b (mod n)
and we say that a is equivalent to b modulo n or that
a is congruent to b modulo n.
In other words a  b (mod n) if a and b have the same
remainder when divided by n.
Also a  b (mod n) if and only if n is a divisor of b-a.

40. relation in Z and produces a partitioned set called
Example  (mod 4)
 (mod n)
defines a equivalence
relation in Z and produces a
partitioned set called
Zn= Z/n ={0,1,2,…,n-1}
in which can be defined
arithmetic operations
a+b (mod n)
a*b (mod n)        
-12
-11
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11
Z/4={[0],[1],[2],[3]}={0,1,2,3}
4+1 (mod 4) = 1
5*2 (mod 4) = 2 12 13 14 15    
[0]
[1]
[2]
[3]

41. Polynomials Given a no negative integer d, a polynomial in n of
degree d is a function p(n) of the form:
Where a1 ,a2 ,…, ad are the coefficients and ad  0.

42. f(n)=O(nd) for some constant d.
A polynomial p(n) is asymptotically positive
if and only if ad > 0.
If p(n), of degree d, is asymptotically positive, we have p(n)= (nd).
a  R , a  0, na is monotonically increasing.
a  R , a  0, na is monotonically decreasing.
A function f(n) is polynomially bounded if
f(n)=O(nd) for some constant d.

43. Exponentials
For all reals a  0, m and n, we have the following identities
a0 = 1
a1 = a
a -1 = 1/a
(am)n = amn
(am)n = (an)m
am an=am+n

44. If a  0 and for all n,
an is monotonically increasing.
a,b  R , a >1,
then nd = o(an)

45. x  R, ex  1+x, equality holds for x=0.
If |x|  1 1+x  ex  1+x+ x2 .
When x 0, ex = 1+x +O(x2) .

46. Logarithms Notations: lg n = log2 n ln n = loge n lgk n = (lg n ) k
lg lg n = lg (lg n )
Logarithms function will only apply to next term in the
formula
lg n + k = (lg n ) +k.
For b > 1 constant and n > 0,
logb n
is strictly increasing.

47. For all reals a > 0, b > 0, c > 0 and n we have the following identities
logc (ab) = logc a + logc b.
logb an = n logb a.
logb (1/an) = - logb a.

48. when x > -1
For x > -1
equality holds for x=0.

49. A function f(n) is polylogaritmically bounded if
f(n)=O(lgk n) for some constant k.
We have the following relation between polynomials
and polylogarithms:
then lgk n = o(nk )

50. Factorials Definition (no recursive) (recursive) Weak upper bound
n!  nn

51. Stirling´s approximation
then
n! = o(nn)
n! =  (2n)
lg(n!) = (n lgn)

52. where

53. Functional iteration
Given a function f (n) the i-th functional iteration of f is
defined as :
with I the identity function. For a particular n, we have,

54. Examples:
f(n) = 2n then
f(n) = n2 then

55. f(n) = nn then

56. Iterated logarithm The iterated logarithm of n, denoted lg* n
“log star of n”is defined as
lg* n , is a very slowly growing function
lg* 1 = 0
lg* 2 = 1
lg* 4 = 2
lg* 16 = 3
lg* = 4
lg* (65536)2 = 5

57. In general k