一. 算法时间复杂度比较符号

 

Definition of Asymptotic Signature

　　一般讨论的 “计算算法的(时间)复杂度” 是人们为了衡量算法性能创造的一种评估方法，我们引入O，Ω，Θ符号来表示算法的复杂度比较关系，在实际分析中我们默认：f,g are monotonic functions and maps from positive to positive.

 

"O" signature：

• • We say f(n) = O(g(n)) if g(n) eventually domains f(n).

　　Or:

• • If there exists a constant c such thta for all sufficient large n: f(n) ≤ c*g(n).　

"Ω" signature:

• • We say f(n) = Ω(g(n)) if f(n) eventually domains g(n).

　　Or:

• • If there exists a constant c such that for all sufficient large n: f(n) ≥ c*g(n).

"Θ" signature:

　　-We say f(n) = Θ(g(n)) if f(n) = O(g(n))   and f(n) = Ω(g(n))

Note here in the definition of big o and big sigma, the constant c locate at the same position at the formular, but we can understand it like this:

• f(n) ≥ c*g(n)
• 1/c*f(n) ≥ g(n)
• a*f(n) ≥ g(n)

so we can think we are searching for a big constant c(a) to magnitude the domianing function in both asymptotic relation.

 

Approach of Sorting Functions in Asymptotic Order

Basically we can classify functions into three types: logarithmic, polinomial, exponencial function.

This is because functions belong to these three types are strictly in ascending asymptotic order:

证明对数函数增长慢于多项式函数：

• • 令f(n) = n, 则 f(n) = O(P(n))
  • 现证log(n) = O(f(n)):

令Φ(n) = c*f(n) - log(n)

令Φ'(n) = c - 1/n = 0, 得c = 1/n

故Φ(n)在 n = 1/c处取最小值为Φ(1/c) = 1 + log(c)

得到：当1 + log(c)>0 时，c*f(n) > log(n) 恒成立

证明指数函数增长快于多项式函数：

The properties of logarithic function:

1. 1. 和差
   2. 换底
   3. 指系
   4. 还原
   5. 互换
   6. 倒数
   7. 链式

Practice：

sorting functions below in ascending asymptotic order:

1. log(nⁿ), n², n^log(n), nlog(log(n)), 2^log(n), log²(n), n^{2^1/2}
2. n^2.5, (2n)^1/2, n+10, 10ⁿ, 100ⁿ, n²logn
3. 2^(logn)1/2, 2ⁿ, n^4/3, n^logn, 2^{2^n}, 2^{n^2}