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hello-algo/docs/chapter_sorting/counting_sort.md

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Add the section of counting sort. (#427)
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# 计数排序
Use underline format for the technical terms (#1213) * Use underline format for the technical terms * Bug fixes
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<u>计数排序（counting sort）</u>通过统计元素数量来实现排序，通常应用于整数数组。
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## 简单实现
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先来看一个简单的例子。给定一个长度为 $n$ 的数组 `nums` ，其中的元素都是“非负整数”，计数排序的整体流程如下图所示。
Add the section of counting sort. (#427)
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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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1. 遍历数组，找出其中的最大数字，记为 $m$ ，然后创建一个长度为 $m + 1$ 的辅助数组 `counter` 。
Polish the chapter of searching and sorting.
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2. **借助 `counter` 统计 `nums` 中各数字的出现次数**，其中 `counter[num]` 对应数字 `num` 的出现次数。统计方法很简单，只需遍历 `nums`（设当前数字为 `num`），每轮将 `counter[num]` 增加 $1$ 即可。
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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3. **由于 `counter` 的各个索引天然有序，因此相当于所有数字已经排序好了**。接下来，我们遍历 `counter` ，根据各数字出现次数从小到大的顺序填入 `nums` 即可。
Add the section of counting sort. (#427)
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Update counting_sort.md
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![计数排序流程](counting_sort.assets/counting_sort_overview.png)
Add the section of counting sort. (#427)
2023-03-19 23:42:12 +08:00
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
2023-12-02 06:21:34 +08:00
代码如下所示：
Squash the language code blocks and fix list.md (#865)
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```src
[file]{counting_sort}-[class]{}-[func]{counting_sort_naive}
```
Add the section of counting sort. (#427)
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Add the section of radix sort. (#441)
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!!! note "计数排序与桶排序的联系"
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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从桶排序的角度看，我们可以将计数排序中的计数数组 `counter` 的每个索引视为一个桶，将统计数量的过程看作将各个元素分配到对应的桶中。本质上，计数排序是桶排序在整型数据下的一个特例。
Add the section of radix sort. (#441)
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Add the section of counting sort. (#427)
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## 完整实现
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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细心的读者可能发现了，**如果输入数据是对象，上述步骤 `3.` 就失效了**。假设输入数据是商品对象，我们想按照商品价格（类的成员变量）对商品进行排序，而上述算法只能给出价格的排序结果。
Add the section of counting sort. (#427)
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Replace "：" with "。"
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那么如何才能得到原数据的排序结果呢？我们首先计算 `counter` 的“前缀和”。顾名思义，索引 `i` 处的前缀和 `prefix[i]` 等于数组前 `i` 个元素之和：
Add the section of counting sort. (#427)
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$$
\text{prefix}[i] = \sum_{j=0}^i \text{counter[j]}
$$
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**前缀和具有明确的意义，`prefix[num] - 1` 代表元素 `num` 在结果数组 `res` 中最后一次出现的索引**。这个信息非常关键，因为它告诉我们各个元素应该出现在结果数组的哪个位置。接下来，我们倒序遍历原数组 `nums` 的每个元素 `num` ，在每轮迭代中执行以下两步。
Add the section of counting sort. (#427)
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1. 将 `num` 填入数组 `res` 的索引 `prefix[num] - 1` 处。
2. 令前缀和 `prefix[num]` 减小 $1$ ，从而得到下次放置 `num` 的索引。
Add the section of counting sort. (#427)
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Mention figures and tables in normal texts. Fix some figures. Finetune texts.
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遍历完成后，数组 `res` 中就是排序好的结果，最后使用 `res` 覆盖原数组 `nums` 即可。下图展示了完整的计数排序流程。
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=== "<1>"
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![计数排序步骤](counting_sort.assets/counting_sort_step1.png)
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=== "<2>"
![counting_sort_step2](counting_sort.assets/counting_sort_step2.png)
=== "<3>"
![counting_sort_step3](counting_sort.assets/counting_sort_step3.png)
=== "<4>"
![counting_sort_step4](counting_sort.assets/counting_sort_step4.png)
=== "<5>"
![counting_sort_step5](counting_sort.assets/counting_sort_step5.png)
=== "<6>"
![counting_sort_step6](counting_sort.assets/counting_sort_step6.png)
=== "<7>"
![counting_sort_step7](counting_sort.assets/counting_sort_step7.png)
=== "<8>"
![counting_sort_step8](counting_sort.assets/counting_sort_step8.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
2023-12-02 06:21:34 +08:00
计数排序的实现代码如下所示：
Add the section of counting sort. (#427)
2023-03-19 23:42:12 +08:00
Squash the language code blocks and fix list.md (#865)
2023-10-17 01:06:00 +08:00
```src
[file]{counting_sort}-[class]{}-[func]{counting_sort}
```
Add the section of counting sort. (#427)
2023-03-19 23:42:12 +08:00
## 算法特性
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
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- **时间复杂度为 $O(n + m)$** ：涉及遍历 `nums` 和遍历 `counter` ，都使用线性时间。一般情况下 $n \gg m$ ，时间复杂度趋于 $O(n)$ 。
- **空间复杂度为 $O(n + m)$、非原地排序**：借助了长度分别为 $n$ 和 $m$ 的数组 `res` 和 `counter` 。
Update the sorting algorithms.
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- **稳定排序**：由于向 `res` 中填充元素的顺序是“从右向左”的，因此倒序遍历 `nums` 可以避免改变相等元素之间的相对位置，从而实现稳定排序。实际上，正序遍历 `nums` 也可以得到正确的排序结果，但结果是非稳定的。
Add the section of counting sort. (#427)
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## 局限性
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
2023-12-02 06:21:34 +08:00
看到这里，你也许会觉得计数排序非常巧妙，仅通过统计数量就可以实现高效的排序。然而，使用计数排序的前置条件相对较为严格。
Add the section of counting sort. (#427)
2023-03-19 23:42:12 +08:00
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
2023-12-02 06:21:34 +08:00
**计数排序只适用于非负整数**。若想将其用于其他类型的数据，需要确保这些数据可以转换为非负整数，并且在转换过程中不能改变各个元素之间的相对大小关系。例如，对于包含负数的整数数组，可以先给所有数字加上一个常数，将全部数字转化为正数，排序完成后再转换回去。
Add the section of counting sort. (#427)
2023-03-19 23:42:12 +08:00
Polish the chapter of searching and sorting.
2023-04-10 03:11:49 +08:00
**计数排序适用于数据量大但数据范围较小的情况**。比如，在上述示例中 $m$ 不能太大，否则会占用过多空间。而当 $n \ll m$ 时，计数排序使用 $O(m)$ 时间，可能比 $O(n \log n)$ 的排序算法还要慢。
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