1. 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.

2. 3.1 Asymptotic notation Θ-notation
 g(n) is an asymptotic tight bound for f(n).
``=’’ abuse
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3. The definition of requires that every member be asymptotically nonnegative.
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4. EXAMPLE:
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5. Why ?
In general,
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6. O-notation (big –oh; Asymptotic Upper Bound)
EXAMPLE: 2n2= O(n3)
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7. Ω-notation (big –omega; Asymptotic Lower Bound)
EXAMPLE:
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8. Theorem 3.1 For any two functions f(n) and g(n), if and only if and .
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9. O-notation (little-oh)
An upper bound that is not asymptotically tight .
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10. ω-notation (little-omega)
An lower bound that is not asymptotically tight .
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11. Relational properties
Transitivity
Reflexivity
Symmetry
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12. Transpose symmetry
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13. 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
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14. 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).
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15. Floor and ceiling
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16. 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.
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17. 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 .
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18. Exponentials
Any positive exponential function with a base greater than 1 grows faster than any polynomial function.
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19. 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.
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20. Factorials
Stirling’s approximation
where
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21. Function iteration
For example, if , then
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22. The iterative logarithm function
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23. 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
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24. Fibonacci numbers
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25. Homework-1 Problem 3-1, 3-2 Due: 10/5/2007 20070928 chap03
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