算法学习之“Big Oh Notation”

最新推荐文章于 2022-12-09 12:54:27 发布

Sagittarius_Warrior

最新推荐文章于 2022-12-09 12:54:27 发布

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分类专栏： Algorithm 文章标签：算法

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大O记法是描述算法渐进复杂度的工具，用于比较算法在问题规模n较大时的增长趋势。它由P. Bachmann在1892年发明。记法中的三个关键元素是起始点n0、倍数c和基准函数。常见的基准函数包括对数函数和线性函数。大O记法具有和、积的数学特性，并且在多项式情况下，通常忽略低阶项和系数。紧致的大O边界是寻找最精确的渐进界，常去除非主导项和常数系数，如将O(n^2+100)简化为O(n^2)，O(10n)简化为O(n)。

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一、Asymptotic analysis

Suppose we are considering two algorithms, A and B, for solving a given problem. Furthermore, let us say that we have done a careful analysis of the running times of each of the algorithms and determined them to be and , respectively, where n is a measure of the problem size. Then it should be a fairly simple matter to compare the two functions and to determine which algorithm is the best!

这段话的意识，比较两种算法的时间成本来判定哪种算法更好。但实际应用中，T(n)是随着问题的规模n变化的，在事先不知道n的情况下，没法比较两种算法的优劣。由此，引入“Asymptotic behavior”，比较两种算法的T(n)在n比较大时的“增长级数”，例如：对数增长、线性增长、指数级增长。

二、Big Oh notation

In 1892, P. Bachmann invented a notation for characterizing the asymptotic behavior of functions. His invention has come to be known as big oh notation:

Definition (Big Oh) Consider a function f( n) which is non-negative for all integers . We say that `` f( n) is big oh g( n),'' which we write f( n)= O( g( n)), if there exists an integer