Chapter 3 Growth of Functions Lee, Hsiu-Hui Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the web.

3.1 Asymptotic notation Θ-notation  g(n) is an asymptotic tight bound for f(n). ``=’’ abuse 20070928 chap03 Hsiu-Hui Lee

The definition of requires that every member be asymptotically nonnegative. 20070928 chap03 Hsiu-Hui Lee

EXAMPLE: 20070928 chap03 Hsiu-Hui Lee

Why ? In general, 20070928 chap03 Hsiu-Hui Lee

O-notation (big –oh; Asymptotic Upper Bound) EXAMPLE: 2n2= O(n3) 20070928 chap03 Hsiu-Hui Lee

Ω-notation (big –omega; Asymptotic Lower Bound) EXAMPLE: 20070928 chap03 Hsiu-Hui Lee

Theorem 3.1 For any two functions f(n) and g(n), if and only if and . 20070928 chap03 Hsiu-Hui Lee

O-notation (little-oh) An upper bound that is not asymptotically tight . 20070928 chap03 Hsiu-Hui Lee

ω-notation (little-omega) An lower bound that is not asymptotically tight . 20070928 chap03 Hsiu-Hui Lee

Relational properties Transitivity Reflexivity Symmetry 20070928 chap03 Hsiu-Hui Lee

Transpose symmetry 20070928 chap03 Hsiu-Hui Lee

Trichotomy Although any two real numbers can be compared, not all functions are asymptotically comparable. a < b, a = b, or a > b. It may be the case that neither nor holds. e.g., are not comparable 20070928 chap03 Hsiu-Hui Lee

3.2 Standard notations and common functions Monotonicity: A function f is monotonically increasing if m  n implies f(m)  f(n). A function f is monotonically decreasing if m  n implies f(m)  f(n). A function f is strictly increasing if m < n implies f(m) < f(n). A function f is strictly decreasing if m > n implies f(m) > f(n). 20070928 chap03 Hsiu-Hui Lee

Floor and ceiling 20070928 chap03 Hsiu-Hui Lee

Modular arithmetic For any integer a and any positive integer n, the value a mod n is the remainder (or residue) of the quotient a/n : a mod n =a -a/nn. If (a mod n) = (b mod n). We write a  b (mod n) and say that a is equivalent to b, modulo n. We write a ≢ b (mod n) if a is not equivalent to b modulo n. 20070928 chap03 Hsiu-Hui Lee

Polynomials Polynomial in n of degree d ≧ 0 If a≧ 0, is monotonically increasing. If a≦0, is monotonically decreasing. A function is polynomial bounded if for some constant k . 20070928 chap03 Hsiu-Hui Lee

Exponentials Any positive exponential function with a base greater than 1 grows faster than any polynomial function. 20070928 chap03 Hsiu-Hui Lee

Logarithms A function f(n) is polylogarithmically bounded if   A function f(n) is polylogarithmically bounded if for any constant a > 0. Any positive polynomial function grows faster than any polylogarithmic function. 20070928 chap03 Hsiu-Hui Lee

Factorials Stirling’s approximation   where 20070928 chap03 Hsiu-Hui Lee

Function iteration For example, if , then 20070928 chap03 Hsiu-Hui Lee

The iterative logarithm function 20070928 chap03 Hsiu-Hui Lee

Since the number of atoms in the observable universe is estimated to be about , which is much less than , we rarely encounter a value of n such that . 20070928 chap03 Hsiu-Hui Lee

Fibonacci numbers 20070928 chap03 Hsiu-Hui Lee

Homework-1 Problem 3-1, 3-2 Due: 10/5/2007 20070928 chap03 Hsiu-Hui Lee