先决条件 – 渐近符号，渐近符号的属性
1.大哦符号 (O) :
大 oh 符号用于描述渐近上限。
在数学上，如果 f(n) 描述算法的运行时间； f(n) 是 O(g(n)) 如果存在正常数 C 并且不存在这样的
0 <=f(n) <= c g(n) for all n>=n0 n = 用于给出a函数的上限。
如果一个函数是 O(n)，那么它也自动是 O(n-square) ！
Big oh (O) 的图形示例：
2. 大欧米茄符号(Ω)：
就像 O 符号提供渐近上限一样，Ω 符号提供渐近下界。
让 f(n) 定义算法的运行时间；
如果存在正常数 C 和 (n 0 ) 使得 f(n) 称为 Ω(g (n))
O<= C g(n) <= f(n) for all n>=n 0 n= 用于给出函数的下界
如果一个函数是 O(n-square)，那么它也自动是 O(n)。
Big Omega (Ω) 的图形示例：
3.大Theta符号(Θ)：
让 f(n) 定义算法的运行时间。
如果 f(n) 为 O(g(n)) 且 f(n) 为 Ω(g(n))，则称 f(n) 为 Θ(g(n))
在数学上，
O<=f(n)<=C 1 g(n) for n>=n 0
O<= C 2 g(n)<=f(n) for n >=n 0合并两个方程，我们得到：
O<=C 2 g(n)<=f(n)<=C 1 g(n) for n>=n 0该等式仅表示存在正常数 C 1 和 C 2 使得 f(n) 夹在 C 2 g(n) 和 C 1 g(n) 之间。
Big Theta (Θ) 的图形示例：
Big oh、Big Omega 和 Big Theta 之间的区别：
| S.No. |
Big Oh |
Big Omega |
Big Theta |
|---|---|---|---|
| 1. | It is like <= rate of growth of an algorithm is less than or equal to a specific value |
It is like >= rate of growth is greater than or equal to a specified value |
It is like == meaning the rate of growth is equal to a specified value |
| 2. | The upper bound of algorithm is represented by Big O notation. Only the above function is bounded by Big O. asymptotic upper bond is it given by Big O notation. | The algorithm’s lower bound is represented by Omega notation. The asymptotic lower bond is given by Omega notation | The bounding of function from above and below is represented by theta notation. The exact asymptotic behavior is done by this theta notation. |
| 3. | Big oh (O) – Worst case | Big Omega (Ω) – Best case | Big Theta (Θ) – Average case |
| 4. | Big-O is a measure of the longest amount of time it could possibly take for the algorithm to complete. | Big- Ω is take a small amount of time as compare to Big-O it could possibly take for the algorithm to complete. | Big- Θ is take very short amount of time as compare to Big-O and Big-? it could possibly take for the algorithm to complete. |
| 5. | Mathematically – Big Oh is 0 <=f(n) <= c g(n) for all n>=n0 | Mathematically – Big Omega is O<= C g(n) <= f(n) for all n>=n 0 | Mathematically – Big Theta is O<=C 2 g(n)<=f(n)<=C 1 g(n) for n>=n 0 |