From ceeb138487881b7488efa06609064534c29c628e Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Sun, 19 Mar 2023 23:42:12 +0800 Subject: [PATCH] Add the section of counting sort. (#427) --- codes/java/chapter_sorting/counting_sort.java | 78 +++++++ .../counting_sort_naive_step1.png | Bin 0 -> 65681 bytes .../counting_sort_naive_step2.png | Bin 0 -> 77046 bytes .../counting_sort_naive_step3.png | Bin 0 -> 64994 bytes .../counting_sort_step1.png | Bin 0 -> 75550 bytes .../counting_sort_step2.png | Bin 0 -> 75504 bytes .../counting_sort_step3.png | Bin 0 -> 88074 bytes .../counting_sort_step4.png | Bin 0 -> 87529 bytes .../counting_sort_step5.png | Bin 0 -> 88375 bytes .../counting_sort_step6.png | Bin 0 -> 89067 bytes .../counting_sort_step7.png | Bin 0 -> 75351 bytes .../counting_sort_step8.png | Bin 0 -> 63572 bytes docs/chapter_sorting/counting_sort.md | 209 ++++++++++++++++++ mkdocs.yml | 5 +- 14 files changed, 290 insertions(+), 2 deletions(-) create mode 100644 codes/java/chapter_sorting/counting_sort.java create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_naive_step1.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_naive_step2.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_naive_step3.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step1.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step2.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step3.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step4.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step5.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step6.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step7.png create mode 100644 docs/chapter_sorting/counting_sort.assets/counting_sort_step8.png create mode 100644 docs/chapter_sorting/counting_sort.md diff --git a/codes/java/chapter_sorting/counting_sort.java b/codes/java/chapter_sorting/counting_sort.java new file mode 100644 index 000000000..79b6292d1 --- /dev/null +++ b/codes/java/chapter_sorting/counting_sort.java @@ -0,0 +1,78 @@ +/** + * File: bubble_sort.java + * Created Time: 2023-03-17 + * Author: Krahets (krahets@163.com) + */ + +package chapter_sorting; + +import java.util.*; + +public class counting_sort { + /* 计数排序 */ + // 简单实现,无法用于排序对象 + static void countingSortNaive(int[] nums) { + // 1. 统计数组最大元素 m + int m = 0; + for (int num : nums) { + m = Math.max(m, num); + } + // 2. 统计各数字的出现次数 + // counter[num] 代表 num 的出现次数 + int[] counter = new int[m + 1]; + for (int num : nums) { + counter[num]++; + } + // 3. 遍历 counter ,将各元素填入原数组 nums + int i = 0; + for (int num = 0; num < m + 1; num++) { + for (int j = 0; j < counter[num]; j++, i++) { + nums[i] = num; + } + } + } + + /* 计数排序 */ + // 完整实现,可排序对象,并且是稳定排序 + static void countingSort(int[] nums) { + // 1. 统计数组最大元素 m + int m = 0; + for (int num : nums) { + m = Math.max(m, num); + } + // 2. 统计各数字的出现次数 + // counter[num] 代表 num 的出现次数 + int[] counter = new int[m + 1]; + for (int num : nums) { + counter[num]++; + } + // 3. 求 counter 的前缀和,将“出现次数”转换为“尾索引” + // 即 counter[num]-1 是 num 在 res 中最后一次出现的索引 + for (int i = 0; i < m; i++) { + counter[i + 1] += counter[i]; + } + // 4. 倒序遍历 nums ,将各元素填入结果数组 res + // 初始化数组 res 用于记录结果 + int n = nums.length; + int[] res = new int[n]; + for (int i = n - 1; i >= 0; i--) { + int num = nums[i]; + res[counter[num] - 1] = num; // 将 num 放置到对应索引处 + counter[num]--; // 令前缀和自减 1 ,得到下次放置 num 的索引 + } + // 使用结果数组 res 覆盖原数组 nums + for (int i = 0; i < n; i++) { + nums[i] = res[i]; + } + } + + public static void main(String[] args) { + int[] nums = { 1, 0, 1, 2, 0, 4, 0, 2, 2, 4 }; + + countingSortNaive(nums); + System.out.println("计数排序(无法排序对象)完成后 nums = " + Arrays.toString(nums)); + + countingSort(nums); + System.out.println("计数排序完成后 nums = " + Arrays.toString(nums)); + } +} diff --git a/docs/chapter_sorting/counting_sort.assets/counting_sort_naive_step1.png 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**基于比较的排序算法**,即通过比较元素之间的大小来实现排序,此类排序算法的时间复杂度无法超越 $O(n \log n)$ 。接下来,我们将学习一种 **非比较排序算法** ,名为「计数排序 Counting Sort」,其时间复杂度可以达到 $O(n)$ 。 + +## 简单实现 + +先看一个简单例子。给定一个长度为 $n$ 的数组 `nums` ,元素皆为 **非负整数**。计数排序的整体流程为: + +1. 统计数组的最大数字,记为 $m$ ,并建立一个长度为 $m + 1$ 的辅助数组 `counter` ; +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]` 自增 $1$ 即可。 +3. **由于 `counter` 的各个索引是天然有序的,因此相当于所有数字已经被排序好了**。接下来,我们遍历 `counter` ,根据各数字的出现次数,将各数字按从小到大的顺序填入 `nums` 即可。 + +=== "<1>" + ![counting_sort_naive_step1](counting_sort.assets/counting_sort_naive_step1.png) + +=== "<2>" + ![counting_sort_naive_step2](counting_sort.assets/counting_sort_naive_step2.png) + +=== "<3>" + ![counting_sort_naive_step3](counting_sort.assets/counting_sort_naive_step3.png) + +以下是实现代码,计数排序名副其实,确实是通过“统计数量”来实现排序的。 + +=== "Java" + + ```java title="counting_sort.java" + [class]{counting_sort}-[func]{countingSortNaive} + ``` + +=== "C++" + + ```cpp title="counting_sort.cpp" + [class]{}-[func]{countingSortNaive} + ``` + +=== "Python" + + ```python title="counting_sort.py" + [class]{}-[func]{counting_sort_naive} + ``` + +=== "Go" + + ```go title="counting_sort.go" + [class]{}-[func]{countingSortNaive} + ``` + +=== "JavaScript" + + ```javascript title="counting_sort.js" + [class]{}-[func]{countingSortNaive} + ``` + +=== "TypeScript" + + ```typescript title="counting_sort.ts" + [class]{}-[func]{countingSortNaive} + ``` + +=== "C" + + ```c title="counting_sort.c" + [class]{}-[func]{countingSortNaive} + ``` + +=== "C#" + + ```csharp title="counting_sort.cs" + [class]{counting_sort}-[func]{countingSortNaive} + ``` + +=== "Swift" + + ```swift title="counting_sort.swift" + [class]{}-[func]{countingSortNaive} + ``` + +=== "Zig" + + ```zig title="counting_sort.zig" + [class]{}-[func]{countingSortNaive} + ``` + +## 完整实现 + +细心的同学可能发现,**如果输入数据是对象,上述步骤 `3.` 就失效了**。例如输入数据是商品对象,我们想要按照商品价格(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。 + +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,即 + +$$ +\text{prefix}[i] = \sum_{j=0}^i \text{counter[j]} +$$ + +**前缀和具有明确意义,`prefix[num] - 1` 代表元素 `num` 在结果数组 `res` 中最后一次出现的索引**。这个信息很关键,因为其给出了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: + +1. 将 `num` 填入数组 `res` 的索引 `prefix[num] - 1` 处; +2. 令前缀和 `prefix[num]` 自减 $1$ ,从而得到下次放置 `num` 的索引; + +完成遍历后,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; + +=== "<1>" + ![counting_sort_step1](counting_sort.assets/counting_sort_step1.png) + +=== "<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) + +计数排序的实现代码如下所示。 + +=== "Java" + + ```java title="counting_sort.java" + [class]{counting_sort}-[func]{countingSort} + ``` + +=== "C++" + + ```cpp title="counting_sort.cpp" + [class]{}-[func]{countingSort} + ``` + +=== "Python" + + ```python title="counting_sort.py" + [class]{}-[func]{counting_sort} + ``` + +=== "Go" + + ```go title="counting_sort.go" + [class]{}-[func]{countingSort} + ``` + +=== "JavaScript" + + ```javascript title="counting_sort.js" + [class]{}-[func]{countingSort} + ``` + +=== "TypeScript" + + ```typescript title="counting_sort.ts" + [class]{}-[func]{countingSort} + ``` + +=== "C" + + ```c title="counting_sort.c" + [class]{}-[func]{countingSort} + ``` + +=== "C#" + + ```csharp title="counting_sort.cs" + [class]{counting_sort}-[func]{countingSort} + ``` + +=== "Swift" + + ```swift title="counting_sort.swift" + [class]{}-[func]{countingSort} + ``` + +=== "Zig" + + ```zig title="counting_sort.zig" + [class]{}-[func]{countingSort} + ``` + +## 算法特性 + +**时间复杂度 $O(n + m)$** :涉及遍历 `nums` 和遍历 `counter` ,都使用线性时间。一般情况下 $n \gg m$ ,此时使用线性 $O(n)$ 时间。 + +**空间复杂度 $O(n + m)$** :数组 `res` 和 `counter` 长度分别为 $n$ , $m$ 。 + +**非原地排序**:借助了辅助数组 `counter` 和结果数组 `res` 的额外空间。 + +**稳定排序**:倒序遍历 `nums` 保持了相等元素的相对位置。 + +**非自适应排序**:与元素分布无关。 + +!!! question "为什么是稳定排序?" + + 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”;其实正序遍历 `nums` 也可以得到正确的排序结果,但结果“非稳定”。 + +## 局限性 + +看到这里,你也许会觉得计数排序太妙了,咔咔一通操作,时间复杂度就下来了。但实际上与其它算法一样,计数排序也无法摆脱“此消彼长”的宿命,**时间复杂度优化的代价是通用型变差**。 + +**计数排序只适用于非负整数**。若想要用在其他类型数据上,则要求该数据必须可以被转化为非负整数,并且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即可。 + +**计数排序只适用于数据范围不大的情况**。比如,上述示例中 $m$ 不能太大,否则占用空间太多;而当 $n \ll m$ 时,计数排序使用 $O(m)$ 时间,有可能比 $O(n \log n)$ 的排序算法还要慢。 diff --git a/mkdocs.yml b/mkdocs.yml index 3c53d1826..c8492e749 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -169,7 +169,7 @@ nav: - 9.     图: - 9.1.   图: chapter_graph/graph.md - 9.2.   图基础操作: chapter_graph/graph_operations.md - - 9.3.   图的遍历: chapter_graph/graph_traversal.md + - 9.3.   图的遍历(New): chapter_graph/graph_traversal.md - 9.4.   小结: chapter_graph/summary.md - 10.     查找算法: - 10.1.   线性查找: chapter_searching/linear_search.md @@ -182,7 +182,8 @@ nav: - 11.3.   插入排序: chapter_sorting/insertion_sort.md - 11.4.   快速排序: chapter_sorting/quick_sort.md - 11.5.   归并排序: chapter_sorting/merge_sort.md - - 11.6.   小结: chapter_sorting/summary.md + - 11.6.   计数排序(New): chapter_sorting/counting_sort.md + - 11.7.   小结: chapter_sorting/summary.md - 12.     附录: - 12.1.   编程环境安装: chapter_appendix/installation.md - 12.2.   一起参与创作: chapter_appendix/contribution.md