Some bug fixes and improvements (#1063)

* Prevent the line breaks of buttons on Safari

* Fix a math symbol

* Small improvements

* Bug fixes and improvements.
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Yudong Jin 2024-01-25 12:50:53 +08:00 committed by GitHub
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<div class="admonition quote"> <div class="admonition quote">
<p align="center">"Knowledge increases by sharing."</p> <p align="center">"Learning by teaching."</p>
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--- ---

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对数阶常见于分治算法。例如归并排序,输入长度为 $n$ 的数组,每轮递归将数组从中点处划分为两半,形成高度为 $\log n$ 的递归树,使用 $O(\log n)$ 栈帧空间。 对数阶常见于分治算法。例如归并排序,输入长度为 $n$ 的数组,每轮递归将数组从中点处划分为两半,形成高度为 $\log n$ 的递归树,使用 $O(\log n)$ 栈帧空间。
再例如将数字转化为字符串,输入一个正整数 $n$ ,它的位数为 $\lfloor \log_{10} n \rfloor + 1$ ,即对应字符串长度为 $\lfloor \log_{10} n \rfloor + 1$ ,因此空间复杂度为 $O(\lfloor \log_{10} n \rfloor + 1) = O(\log n)$ 。 再例如将数字转化为字符串,输入一个正整数 $n$ ,它的位数为 $\lfloor \log_{10} n \rfloor + 1$ ,即对应字符串长度为 $\lfloor \log_{10} n \rfloor + 1$ ,因此空间复杂度为 $O(\log_{10} n + 1) = O(\log n)$ 。
## 权衡时间与空间 ## 权衡时间与空间

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插入排序的时间复杂度为 $O(n^2)$ ,而我们即将学习的快速排序的时间复杂度为 $O(n \log n)$ 。尽管插入排序的时间复杂度更高,**但在数据量较小的情况下,插入排序通常更快**。 插入排序的时间复杂度为 $O(n^2)$ ,而我们即将学习的快速排序的时间复杂度为 $O(n \log n)$ 。尽管插入排序的时间复杂度更高,**但在数据量较小的情况下,插入排序通常更快**。
这个结论与线性查找和二分查找的适用情况的结论类似。快速排序这类 $O(n \log n)$ 的算法属于基于分治策略的排序算法,往往包含更多单元计算操作。而在数据量较小时,$n^2$ 和 $n \log n$ 的数值比较接近,复杂度不占主导地位每轮中的单元操作数量起到决定性作用。 这个结论与线性查找和二分查找的适用情况的结论类似。快速排序这类 $O(n \log n)$ 的算法属于基于分治策略的排序算法,往往包含更多单元计算操作。而在数据量较小时,$n^2$ 和 $n \log n$ 的数值比较接近,复杂度不占主导地位每轮中的单元操作数量起到决定性作用。
实际上,许多编程语言(例如 Java的内置排序函数采用了插入排序大致思路为对于长数组采用基于分治策略的排序算法例如快速排序对于短数组直接使用插入排序。 实际上,许多编程语言(例如 Java的内置排序函数采用了插入排序大致思路为对于长数组采用基于分治策略的排序算法例如快速排序对于短数组直接使用插入排序。

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<div class="section-content"> <div class="section-content">
<h3 style="text-align: center; margin: 1em auto;">推荐语</h3> <h3 style="text-align: center; margin: 1em auto;">推荐语</h3>
<div class="intro-container" style="margin-bottom: 1em;"> <div class="intro-container" style="margin: 0 auto;">
<div class="intro-text endor-text"> <div class="intro-text endor-text">
<p style="margin-bottom: 0;">“一本通俗易懂的数据结构与算法入门书,引导读者手脑并用地学习,强烈推荐算法初学者阅读。”</p> <p style="margin-bottom: 0;">“一本通俗易懂的数据结构与算法入门书,引导读者手脑并用地学习,强烈推荐算法初学者阅读。”</p>
<p style="font-weight: bold;">—— 邓俊辉,清华大学计算机系教授</p> <p style="font-weight: bold;">—— 邓俊辉,清华大学计算机系教授</p>
@ -250,7 +250,7 @@ hide:
<h3 style="margin-left: 0.5em;">互助学习</h3> <h3 style="margin-left: 0.5em;">互助学习</h3>
</div> </div>
<p style="margin: 0;">欢迎讨论与提问,读者间携手共进</p> <p style="margin: 0;">欢迎讨论与提问,读者间携手共进</p>
<p class="intro-quote">"Knowledge increases by sharing."</br>“知识在分享中得以增长”</p> <p class="intro-quote">"Learning by teaching."</br>“教学相长”</p>
</div> </div>
</div> </div>
<img class="intro-image" src="index.assets/comment.gif"> <img class="intro-image" src="index.assets/comment.gif">

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padding: 2em; padding: 3em 2em;
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@ -221,7 +221,7 @@ body {
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