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3.G rowth of Functions. 3.1 Asymptotic notation .  g ( n ) is an asymptotic tight bound for f ( n ). ``=’’ abuse. The definition of required every member of be asymptotically nonnegative. Example: . In general, . asymptotic upper bound. asymptotic lower bound . Theorem 3.1.

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3 g rowth of functions

3.Growth of Functions


3 1 asymptotic notation
3.1 Asymptotic notation

g(n) is an asymptotic tight bound for f(n).

  • ``=’’ abuse



Example
Example: asymptotically nonnegative.

  • In general,


Asymptotic upper bound
asymptotic upper bound asymptotically nonnegative.


Asymptotic lower bound
asymptotic lower bound asymptotically nonnegative.


Theorem 3 1
Theorem 3.1. asymptotically nonnegative.

  • For any two functions f(n) and g(n), if and only if and




Trichotomy
Trichotomy asymptotically nonnegative.

  • a < b, a = b, or a > b.

  • e.g., does not always holdbecause the latter oscillates between


2 2 standard notations and common functions
2.2 Standard notations and common functions asymptotically nonnegative.

  • 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).


Floor and ceiling
Floor and ceiling asymptotically nonnegative.

For any real x

For any integer n

For any real n>=0, and

integers a, b >0


Proof of ceiling
Proof of Ceiling asymptotically nonnegative.


Proof of floor
Proof of Floor asymptotically nonnegative.


Modular arithmetic
Modular arithmetic asymptotically nonnegative.

  • 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.


Exponential basics
Exponential Basics asymptotically nonnegative.


Polynomials v s exponentials
Polynomials asymptotically nonnegative. v.s. Exponentials

  • Polynomials:

    • A function is polynomial bounded if f(n)=O(nk)

  • Exponentials:

    • Any positive exponential function grows faster than any polynomial.


Prove n b o a n
Prove: asymptotically nonnegative.nb=o(an)


Logarithm notations
Logarithm Notations asymptotically nonnegative.


Logarithm basics
Logarithm Basics asymptotically nonnegative.


Logarithms
Logarithms asymptotically nonnegative.

  • A function f(n) is polylogarithmically bounded if

  • for any constant a > 0.

  • Any positive polynomial function grows faster than any polylogarithmic function.


Prove lg b n o n a
Prove: asymptotically nonnegative.lgbn=o(na)


Factorials
Factorials asymptotically nonnegative.

  • Stirling’s approximation

where


Function iteration
Function iteration asymptotically nonnegative.

For example, if , then


The iterative logarithm function
The iterative logarithm function asymptotically nonnegative.

h



Fibonacci numbers
Fibonacci numbers estimated to be about , which is much less than , we rarely encounter a value of


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