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Complexity/复杂度

24/10/2021 KevinZonda

Big-O notation

$$ f(n)=O(g(n))\Longleftrightarrow|f(n)|\leq|cg(n)| \\ \text{for positive constants } c, n_0 \text{ where } n \gt n_0. $$

It is a UPPER BOUND.

Example: $$ 2n = O(n)\Longleftrightarrow |2n|\leq|cn| \ 3n^2+n=O(n^2)\Longleftrightarrow |3n^2+n|\leq|cn^2| $$ Big O is a notation not a function, thus O(f(n)) is not a function.

Example: $3n^2+4=O(n^2)$ is just a shorthand way pf writing $|3n^2+4|\leq|cn^2)|$ for some constant C.

$O(n^2)=3n^2+4$ has no meaning.

Big O notation can be made mathematically precise by defining it to be the class of functions with complexity O(f (n)). Therefore we can say that the complexity of an algorithm is "in" O(f (n)) or, shortening it, simply say that, for an algorithm X, we have that $X \in O(f(n))$.

大 O 符号为上界其存在逻辑为 $$ \forall n \geq n_0, 0 \leq f(n) \leq cg(n) $$

Other Measure Ways

  • Big O: $f(n)=O(g(n))$: g is an upper bound on how fast $f$ grows as $n$ increases.
  • Little o: $f(n)=o(g(n))$: A stricter upper bound than Big O.
  • Theta: $f(n)=\Theta(g(n))$: More precise than Big O and Little o, it provides both upper and lower bounds, which are given by the same function, except with different constant factors.
    That is, $f$ and $g$ grow at the same rate.
  • Asymptotically Equal: $f(n)∼g(n)$: stricter upper and lower bounds
  • Omega: $f(n)=\Omega (g(n))$: an absolute lower bound (the negation of little o)

Little o

$$ f(n)=o(g(n))\Longleftrightarrow \lim_{n\rightarrow\infty}{\cfrac{f(n)}{g(n)}} \ \text{exists and is equal to 0.} $$

Example: $$ 2n^2=O(n^3) \ \text{and} \ 2n^2=O(n^2) $$

$$ 2n^2 =o(n^3)\\ \because\lim_{n\rightarrow\infty}{\cfrac{2n^2}{n^3}}=\lim_{n\rightarrow\infty}{\cfrac{2}{n}}=0\\ \text{Not} \ 2n^2=o(n^2)\\ \because\lim_{n\rightarrow\infty}{\cfrac{2n^2}{n^2}}=\lim_{n\rightarrow\infty}{2}=2\ne0\\ \text{Not} \ n^2=o(n^2)\\ \because\lim_{n\rightarrow\infty}{\cfrac{n^2}{n^2}}=\lim_{n\rightarrow\infty}{1}=1\ne0\\ $$

Theta

$$ f(n)=\Theta(g(n))\Longleftrightarrow c_ig(n) \leq f(n)\leq c_2g(n)\\ \text{positive constants}\ c_1, c_2, n_0 \text{ and } n \gt n_o $$

也就是说，我们可以找到两个正数 $c_1, c_2$ 把 $f(n)$ 夹在 $c_1g(n)$ 与 $c_2g(n)$ 中间。

Asymptotically Equal/渐进相等

$$ f(n)\sim g(n) \Longleftrightarrow\lim_{n\rightarrow\infty}{\cfrac{f(n)}{g(n)}} \ \text{exists and is equal to 1.} $$

也就是说，其在 $n\rightarrow\infty$ 情况下，原函数与渐进函数的比值为 1 。

Omega

$$ f(n)=\Omega(g(n)) \Longleftrightarrow |f(n)| \geq |cg(n)| \\ \text{for positive constants } c, n_0 \text{ where } n \gt n_0. $$

It is a LOWER BOUND.

Little omega

$$ f(n)=\omega(g(n))\Longleftrightarrow \lim_{n\rightarrow\infty}{\cfrac{g(n)}{f(n)}} \ \text{exists and is equal to 0.} $$

Different Kinds of Complexities

  • Average Case complexity
    = average complexity over all possible inputs/situations
    (we need to know the likelihood of each of the input!)
  • Worst Case complexity
    = the worst complexity over all possible inputs/situations
  • Best Case complexity
    = the best complexity over all possible inputs/situations
  • Amortized complexity
    = average time taken over a sequence of consecutive operations

Amortized Complexity

Chinese: 均摊复杂度

References

  1. "复杂度 - OI Wiki" - https://oi-wiki.org/basic/complexity/