diff --git a/codes/cpp/chapter_sorting/CMakeLists.txt b/codes/cpp/chapter_sorting/CMakeLists.txt index d1f11a857..e6347cf9f 100644 --- a/codes/cpp/chapter_sorting/CMakeLists.txt +++ b/codes/cpp/chapter_sorting/CMakeLists.txt @@ -1,4 +1,6 @@ +add_executable(selection_sort selection_sort.cpp) add_executable(bubble_sort bubble_sort.cpp) add_executable(insertion_sort insertion_sort.cpp) add_executable(merge_sort merge_sort.cpp) -add_executable(quick_sort quick_sort.cpp) \ No newline at end of file +add_executable(quick_sort quick_sort.cpp) +add_executable(heap_sort heap_sort.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_sorting/heap_sort.cpp b/codes/cpp/chapter_sorting/heap_sort.cpp new file mode 100644 index 000000000..57af471c0 --- /dev/null +++ b/codes/cpp/chapter_sorting/heap_sort.cpp @@ -0,0 +1,54 @@ +/** + * File: heap_sort.cpp + * Created Time: 2023-05-26 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 堆的长度为 n ，从节点 i 开始，从顶至底堆化 */ +void siftDown(vector &nums, int n, int i) { + while (true) { + // 判断节点 i, l, r 中值最大的节点，记为 ma + int l = 2 * i + 1; + int r = 2 * i + 2; + int ma = i; + if (l < n && nums[l] > nums[ma]) + ma = l; + if (r < n && nums[r] > nums[ma]) + ma = r; + // 若节点 i 最大或索引 l, r 越界，则无需继续堆化，跳出 + if (ma == i) { + break; + } + // 交换两节点 + swap(nums[i], nums[ma]); + // 循环向下堆化 + i = ma; + } +} + +/* 堆排序 */ +void heapSort(vector &nums) { + // 建堆操作：堆化除叶节点以外的其他所有节点 + for (int i = nums.size() / 2 - 1; i >= 0; --i) { + siftDown(nums, nums.size(), i); + } + // 从堆中提取最大元素，循环 n-1 轮 + for (int i = nums.size() - 1; i > 0; --i) { + // 交换根节点与最右叶节点（即交换首元素与尾元素） + swap(nums[0], nums[i]); + // 以根节点为起点，从顶至底进行堆化 + siftDown(nums, i, 0); + } +} + +/* Driver Code */ +int main() { + vector nums = {4, 1, 3, 1, 5, 2}; + heapSort(nums); + cout << "堆排序完成后 nums = "; + printVector(nums); + + return 0; +} diff --git a/codes/java/chapter_sorting/heap_sort.java b/codes/java/chapter_sorting/heap_sort.java new file mode 100644 index 000000000..514d85931 --- /dev/null +++ b/codes/java/chapter_sorting/heap_sort.java @@ -0,0 +1,57 @@ +/** + * File: heap_sort.java + * Created Time: 2023-05-26 + * Author: Krahets (krahets@163.com) + */ + +package chapter_sorting; + +import java.util.Arrays; + +public class heap_sort { + /* 堆的长度为 n ，从节点 i 开始，从顶至底堆化 */ + public static void siftDown(int[] nums, int n, int i) { + while (true) { + // 判断节点 i, l, r 中值最大的节点，记为 ma + int l = 2 * i + 1; + int r = 2 * i + 2; + int ma = i; + if (l < n && nums[l] > nums[ma]) + ma = l; + if (r < n && nums[r] > nums[ma]) + ma = r; + // 若节点 i 最大或索引 l, r 越界，则无需继续堆化，跳出 + if (ma == i) + break; + // 交换两节点 + int temp = nums[i]; + nums[i] = nums[ma]; + nums[ma] = temp; + // 循环向下堆化 + i = ma; + } + } + + /* 堆排序 */ + public static void heapSort(int[] nums) { + // 建堆操作：堆化除叶节点以外的其他所有节点 + for (int i = nums.length / 2 - 1; i >= 0; i--) { + siftDown(nums, nums.length, i); + } + // 从堆中提取最大元素，循环 n-1 轮 + for (int i = nums.length - 1; i > 0; i--) { + // 交换根节点与最右叶节点（即交换首元素与尾元素） + int tmp = nums[0]; + nums[0] = nums[i]; + nums[i] = tmp; + // 以根节点为起点，从顶至底进行堆化 + siftDown(nums, i, 0); + } + } + + public static void main(String[] args) { + int[] nums = { 4, 1, 3, 1, 5, 2 }; + heapSort(nums); + System.out.println("堆排序完成后 nums = " + Arrays.toString(nums)); + } +} diff --git a/codes/python/chapter_sorting/heap_sort.py b/codes/python/chapter_sorting/heap_sort.py new file mode 100644 index 000000000..6c75981aa --- /dev/null +++ b/codes/python/chapter_sorting/heap_sort.py @@ -0,0 +1,45 @@ +""" +File: heap_sort.py +Created Time: 2023-05-24 +Author: Krahets (krahets@163.com) +""" + + +def sift_down(nums: list[int], n: int, i: int): + """堆的长度为 n ，从节点 i 开始，从顶至底堆化""" + while True: + # 判断节点 i, l, r 中值最大的节点，记为 ma + l = 2 * i + 1 + r = 2 * i + 2 + ma = i + if l < n and nums[l] > nums[ma]: + ma = l + if r < n and nums[r] > nums[ma]: + ma = r + # 若节点 i 最大或索引 l, r 越界，则无需继续堆化，跳出 + if ma == i: + break + # 交换两节点 + nums[i], nums[ma] = nums[ma], nums[i] + # 循环向下堆化 + i = ma + + +def heap_sort(nums: list[int]): + """堆排序""" + # 建堆操作：堆化除叶节点以外的其他所有节点 + for i in range(len(nums) // 2 - 1, -1, -1): + sift_down(nums, len(nums), i) + # 从堆中提取最大元素，循环 n-1 轮 + for i in range(len(nums) - 1, 0, -1): + # 交换根节点与最右叶节点（即交换首元素与尾元素） + nums[0], nums[i] = nums[i], nums[0] + # 以根节点为起点，从顶至底进行堆化 + sift_down(nums, i, 0) + + +"""Driver Code""" +if __name__ == "__main__": + nums = [4, 1, 3, 1, 5, 2] + heap_sort(nums) + print("堆排序完成后 nums =", nums) diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step1.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step1.png new file mode 100644 index 000000000..93628563d Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step1.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step10.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step10.png new file mode 100644 index 000000000..4ce322119 Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step10.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step11.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step11.png new file mode 100644 index 000000000..337a9cc49 Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step11.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step12.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step12.png new file mode 100644 index 000000000..465fe404c Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step12.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step2.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step2.png new file mode 100644 index 000000000..13e714f5c Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step2.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step3.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step3.png new file mode 100644 index 000000000..6d0e91b5d Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step3.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step4.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step4.png new file mode 100644 index 000000000..7da1d55b0 Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step4.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step5.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step5.png new file mode 100644 index 000000000..bbbfa6e78 Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step5.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step6.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step6.png new file mode 100644 index 000000000..5390ad38d Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step6.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step7.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step7.png new file mode 100644 index 000000000..4db0c65be Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step7.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step8.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step8.png new file mode 100644 index 000000000..388fb8ad1 Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step8.png differ diff --git a/docs/chapter_sorting/heap_sort.assets/heap_sort_step9.png b/docs/chapter_sorting/heap_sort.assets/heap_sort_step9.png new file mode 100644 index 000000000..43fd42125 Binary files /dev/null and b/docs/chapter_sorting/heap_sort.assets/heap_sort_step9.png differ diff --git a/docs/chapter_sorting/heap_sort.md b/docs/chapter_sorting/heap_sort.md new file mode 100644 index 000000000..0c3a0c7f9 --- /dev/null +++ b/docs/chapter_sorting/heap_sort.md @@ -0,0 +1,144 @@ +# 堆排序 + +!!! tip + + 阅读本节前，请确保已完成堆章节的学习。 + +「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： + +1. 输入数组并建立小顶堆，此时最小元素位于堆顶。 +2. 初始化一个数组 `res` ，用于存储排序结果。 +3. 循环执行 $n$ 轮出堆操作，并依次将出堆元素记录至 `res` ，即可得到从小到大排序的序列。 + +该方法虽然可行，但需要借助一个额外数组，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。设数组的长度为 $n$ ，堆排序的流程如下： + +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。 +4. 循环执行第 `2.` 和 `3.` 步。循环 $n - 1$ 轮后，即可完成数组排序。 + +实际上，元素出堆操作中也包含第 `2.` 和 `3.` 步，只是多了一个弹出元素的步骤。 + +=== "<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) + +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）的函数。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。 + +=== "Java" + + ```java title="heap_sort.java" + [class]{heap_sort}-[func]{siftDown} + + [class]{heap_sort}-[func]{heapSort} + ``` + +=== "C++" + + ```cpp title="heap_sort.cpp" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +=== "Python" + + ```python title="heap_sort.py" + [class]{}-[func]{sift_down} + + [class]{}-[func]{heap_sort} + ``` + +=== "Go" + + ```go title="heap_sort.go" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +=== "JavaScript" + + ```javascript title="heap_sort.js" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +=== "TypeScript" + + ```typescript title="heap_sort.ts" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +=== "C" + + ```c title="heap_sort.c" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +=== "C#" + + ```csharp title="heap_sort.cs" + [class]{heap_sort}-[func]{siftDown} + + [class]{heap_sort}-[func]{heapSort} + ``` + +=== "Swift" + + ```swift title="heap_sort.swift" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +=== "Zig" + + ```zig title="heap_sort.zig" + [class]{}-[func]{siftDown} + + [class]{}-[func]{heapSort} + ``` + +## 算法特性 + +- **时间复杂度 $O(n \log n)$ 、非自适应排序** ：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。 +- **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。 diff --git a/mkdocs.yml b/mkdocs.yml index 589ca3aa1..83d96fcf2 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -181,15 +181,16 @@ nav: - 10.5.   小结: chapter_searching/summary.md - 11.     排序算法: - 11.1.   排序算法: chapter_sorting/sorting_algorithm.md - - 11.2.   选择排序(New): chapter_sorting/selection_sort.md + - 11.2.   选择排序（New）: chapter_sorting/selection_sort.md - 11.3.   冒泡排序: chapter_sorting/bubble_sort.md - 11.4.   插入排序: chapter_sorting/insertion_sort.md - 11.5.   快速排序: chapter_sorting/quick_sort.md - 11.6.   归并排序: chapter_sorting/merge_sort.md - - 11.7.   桶排序: chapter_sorting/bucket_sort.md - - 11.8.   计数排序: chapter_sorting/counting_sort.md - - 11.9.   基数排序: chapter_sorting/radix_sort.md - - 11.10.   小结: chapter_sorting/summary.md + - 11.7.   堆排序（New）: chapter_sorting/heap_sort.md + - 11.8.   桶排序: chapter_sorting/bucket_sort.md + - 11.9.   计数排序: chapter_sorting/counting_sort.md + - 11.10.   基数排序: chapter_sorting/radix_sort.md + - 11.11.   小结: chapter_sorting/summary.md - 12.     回溯算法: - 12.1.   回溯算法（New）: chapter_backtracking/backtracking_algorithm.md - 12.2.   全排列问题（New）: chapter_backtracking/permutations_problem.md