Descripción de las Notaciones asintoticas

1. A A nalisis de lgoritmos Informacion tomada del libro de Cormen.

3. ASYMPTOTIC NOTATION

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

5. Theta   ( g(n) ) = { f : N  R * | (  c 1 , c 2  R + ) (  n 0  N) (  n  n 0 ) ( 0  c 1 g(n)  f(n)  c 2 g(n) ) } c 1  lim f(n)  c 2 n  g(n ) n 0 g c 1 g f c 2 g

7. Example Lets show that We have to find c 1 , c 2 and n 0 such that

8. For all n  n 0 , Dividing by n 2 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 c 2  1/2 . Likewise the left hand inequality can be made to hold for n  7 by choosing c 1  1/14 .

10. Thus by choosing c 1 = 1/14 , c 2 = 1/2 and n 0 =7 then we can verify that

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

13. Example Lets show that We have to find c and n 0 such that

14. For all n  n 0 , 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 n 0 =1 then we can verify that

15. n 0 =1 a+b b a

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

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

20. n 2 n Examples Lets show that We only have to show

21. Lets show that We have 3n n

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

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

33. Asymptotic notation two variables O(g(m, n))= { f: N  N  R * | (  c  R + )(  m 0 , n 0  N) (  n  n 0 ) (  m  m 0 ) (f(m, n)  c g(m, n)) }

34. COMMON FUNCTIONS

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

41. Polynomials Given a no negative integer d, a polynomial in n of degree d is a function p(n) of the form: Where a 1 , a 2 , …, a d are the coefficients and a d  0.

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

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

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) = n 2 then

55. f(n) = n n then

57. In general k