我在问更多关于这对我的代码意味着什么。我能从数学上理解这些概念,只是很难理解它们在概念上的含义。例如,如果有人要对一个数据结构执行O(1)操作,我知道它必须执行的操作数量不会增长,因为有更多的项。O(n)操作意味着您将对每个元素执行一组操作。有人能帮我填一下吗?

比如O(n²)的运算会怎样? 如果一个操作是O(nlog (n))这是什么意思? 有人必须吸可卡因才能写出O(x!)吗?


当前回答

一种思考的方式是:

O(N²)意味着对于每个元素,你都要对其他元素做一些事情,比如比较它们。冒泡排序就是一个例子。

O(N log N)意味着对于每个元素,你只需要看log N个元素。这通常是因为你知道一些元素,可以让你做出有效的选择。最有效的排序就是一个例子,比如归并排序。

O(N!)表示对N个元素的所有可能排列进行处理。旅行推销员就是一个例子,那里有N!访问节点的方法,暴力解决方案是查看每一种可能的排列的总代价,以找到最优的一个。

其他回答

log(n) means logarithmic growth. An example would be divide and conquer algorithms. If you have 1000 sorted numbers in an array ( ex. 3, 10, 34, 244, 1203 ... ) and want to search for a number in the list (find its position), you could start with checking the value of the number at index 500. If it is lower than what you seek, jump to 750. If it is higher than what you seek, jump to 250. Then you repeat the process until you find your value (and key). Every time we jump half the search space, we can cull away testing many other values since we know the number 3004 can't be above number 5000 (remember, it is a sorted list).

N log(N)表示N * log(N)

不,O(n)算法并不意味着它将对每个元素执行操作。大o符号给了你一种方法来谈论你的算法的“速度”独立于你的实际机器。

O(n)表示算法花费的时间随着输入的增加而线性增长。O(n²)意味着你的算法花费的时间是你输入的平方。等等。

我是这样向我那些不懂技术的朋友描述的:

考虑多位数加法。很好的老式铅笔和纸的补充。就是你7-8岁时学的那种。给定两个三位数或四位数,你很容易就能求出它们加起来是多少。

如果我给你两个100位的数字,然后问你它们加起来是多少,即使你必须使用铅笔和纸,计算出来也会非常简单。一个聪明的孩子可以在几分钟内做这样的加法。这只需要大约100次操作。

现在,考虑多位数乘法。你可能在八九岁的时候就学会了。你(希望)做了很多重复的练习来学习它背后的机制。

Now, imagine I gave you those same two 100-digit numbers and told you to multiply them together. This would be a much, much harder task, something that would take you hours to do - and that you'd be unlikely to do without mistakes. The reason for this is that (this version of) multiplication is O(n^2); each digit in the bottom number has to be multiplied by each digit in the top number, leaving a total of about n^2 operations. In the case of the 100-digit numbers, that's 10,000 multiplications.

你可能会发现把它形象化很有用:

同样,在LogY/LogX尺度上,函数n1/2, n, n2都看起来像直线,而在LogY/X尺度上,2n, en, 10n是直线和n!是线性的(看起来像n log n)

只是为了回应我上面帖子的一些评论:

Domenic - I'm on this site, and I care. Not for pedantry's sake, but because we - as programmers - typically care about precision. Using O( ) notation incorrectly in the style that some have done here renders it kind of meaningless; we may just as well say something takes n^2 units of time as O( n^2 ) under the conventions used here. Using the O( ) adds nothing. It's not just a small discrepancy between common usage and mathematical precision that I'm talking about, it's the difference between it being meaningful and it not.

我知道很多很多优秀的程序员都准确地使用这些术语。说“哦,我们是程序员,所以我们不在乎”会降低整个企业的成本。

一个接一个-嗯,不完全是,尽管我同意你的观点。对于任意大的n,它不是O(1)这是O()的定义。它只是表明O()对于有界n的适用性有限,在这里我们更愿意讨论所走的步数,而不是这个数字的界限。