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feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
# 堆排序
!!! tip
1. lower-case nouns 2. fix 2 figures 3. Replace some 「」 by “”
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阅读本节前，请确保已学完“堆“章节。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
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Replace "：" with "。"
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「堆排序 heap sort」是一种基于堆数据结构实现的高效排序算法。我们可以利用已经学过的“建堆操作”和“元素出堆操作”实现堆排序。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
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1. 输入数组并建立小顶堆，此时最小元素位于堆顶。
Polish some texts.
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2. 不断执行出堆操作，依次记录出堆元素，即可得到从小到大排序的序列。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
Polish some texts.
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以上方法虽然可行，但需要借助一个额外数组来保存弹出的元素，比较浪费空间。在实际中，我们通常使用一种更加优雅的实现方式。
## 算法流程
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
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设数组的长度为 $n$ ，堆排序的流程如下图所示。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
1. 输入数组并建立大顶堆。完成后，最大元素位于堆顶。
2. 将堆顶元素（第一个元素）与堆底元素（最后一个元素）交换。完成交换后，堆的长度减 $1$ ，已排序元素数量加 $1$ 。
3. 从堆顶元素开始，从顶到底执行堆化操作（Sift Down）。完成堆化后，堆的性质得到修复。
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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4. 循环执行第 `2.` 步和第 `3.` 步。循环 $n - 1$ 轮后，即可完成数组排序。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
2023-08-21 19:33:45 +08:00
!!! tip
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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实际上，元素出堆操作中也包含第 `2.` 步和第 `3.` 步，只是多了一个弹出元素的步骤。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
=== "<1>"
![堆排序步骤](heap_sort.assets/heap_sort_step1.png)
=== "<2>"
![heap_sort_step2](heap_sort.assets/heap_sort_step2.png)
=== "<3>"
![heap_sort_step3](heap_sort.assets/heap_sort_step3.png)
=== "<4>"
![heap_sort_step4](heap_sort.assets/heap_sort_step4.png)
=== "<5>"
![heap_sort_step5](heap_sort.assets/heap_sort_step5.png)
=== "<6>"
![heap_sort_step6](heap_sort.assets/heap_sort_step6.png)
=== "<7>"
![heap_sort_step7](heap_sort.assets/heap_sort_step7.png)
=== "<8>"
![heap_sort_step8](heap_sort.assets/heap_sort_step8.png)
=== "<9>"
![heap_sort_step9](heap_sort.assets/heap_sort_step9.png)
=== "<10>"
![heap_sort_step10](heap_sort.assets/heap_sort_step10.png)
=== "<11>"
![heap_sort_step11](heap_sort.assets/heap_sort_step11.png)
=== "<12>"
![heap_sort_step12](heap_sort.assets/heap_sort_step12.png)
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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在代码实现中，我们使用了与“堆”章节相同的从顶至底堆化 `sift_down()` 函数。值得注意的是，由于堆的长度会随着提取最大元素而减小，因此我们需要给 `sift_down()` 函数添加一个长度参数 $n$ ，用于指定堆的当前有效长度。代码如下所示：
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
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Squash the language code blocks and fix list.md (#865)
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```src
[file]{heap_sort}-[class]{}-[func]{heap_sort}
```
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
## 算法特性
Update punctuation
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- **时间复杂度 $O(n \log n)$、非自适应排序**：建堆操作使用 $O(n)$ 时间。从堆中提取最大元素的时间复杂度为 $O(\log n)$ ，共循环 $n - 1$ 轮。
- **空间复杂度 $O(1)$、原地排序**：几个指针变量使用 $O(1)$ 空间。元素交换和堆化操作都是在原数组上进行的。
feat: Add the section of heap sort. (#516) * Add the section of heap sort. * Update heap_sort.cpp
2023-05-26 04:46:56 +08:00
- **非稳定排序**：在交换堆顶元素和堆底元素时，相等元素的相对位置可能发生变化。
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