From fdb175311c99216031da290659a1ad67d686f1ed Mon Sep 17 00:00:00 2001 From: fiteen Date: Fri, 27 Dec 2019 17:20:02 +0800 Subject: [PATCH] feat: add RadixSort --- README.md | 17 ++++---- RadixSort/README.md | 90 +++++++++++++++++++++++++++++++++++++++ RadixSort/radix-sort.gif | Bin 0 -> 1401109 bytes RadixSort/radix_sort.c | 60 ++++++++++++++++++++++++++ 4 files changed, 159 insertions(+), 8 deletions(-) create mode 100644 RadixSort/README.md create mode 100644 RadixSort/radix-sort.gif create mode 100644 RadixSort/radix_sort.c diff --git a/README.md b/README.md index 001b832..30377d6 100644 --- a/README.md +++ b/README.md @@ -2,7 +2,7 @@ 排序算法是程序员必备的基础知识,弄明白它们的原理和实现很有必要。 -我们可以将常见的排序算法可以分成两类: +我们可以将常见的**内部排序算法**可以分成两类: ![](sort-category.png) @@ -20,16 +20,17 @@ **非比较类排序**:不通过比较来决定元素间的相对次序,其时间复杂度可以突破 O(nlogn),以线性时间运行。属于非比较类的有: -| 排序算法 | 时间复杂度 | 最差情况 | 最好情况 | 空间复杂度 | 稳定性 | 排序方式 | -| :----------------------: | :--------: | :------: | :------: | :--------: | :----: | :-------: | -| [计数排序](CountingSort) | O(n+k)​ | O(n+k)​ | O(n+k)​ | O(n+k)​ | 稳定 | Out-place | -| [桶排序](BucketSort) | O(n+k)​ | O(n²) | O(n)​ | O(n+m)​ | 稳定 | Out-place | -| 基数排序 | O(n×k)​ | O(n×k)​ | O(n×k)​ | O(n+k)​ | 稳定 | Out-place | - +| 排序算法 | 时间复杂度 | 最差情况 | 最好情况 | 空间复杂度 | 稳定性 | 排序方式 | +| :----------------------: | :--------: | :-------: | :------: | :--------: | :----: | :-------: | +| [桶排序](BucketSort) | O(n+k)​ | O(n²) | O(n)​ | O(n+r)​ | 稳定 | Out-place | +| [计数排序](CountingSort) | O(n+k)​ | O(n+k)​ | O(n+k)​ | O(n+k)​ | 稳定 | Out-place | +| [基数排序](RadixSort) | O(d(n+r))​ | O(d(n+r)) | O(d(n+r)) | O(n+r)​ | 稳定 | Out-place | **n**:待排序列的个数 -**m**:“桶”的个数 +**r**:“桶”的个数 + +**d**:待排序列的最高位数 **In-place**:原地算法,指的是占用常用内存,不占用额外内存。空间复杂度为 O(1) 的都可以认为是原地算法 diff --git a/RadixSort/README.md b/RadixSort/README.md new file mode 100644 index 0000000..9c60d62 --- /dev/null +++ b/RadixSort/README.md @@ -0,0 +1,90 @@ +## 基数排序 + +基数排序(Radix Sort)是**非比较型**排序算法,它和[计数排序](../CountingSort)、[桶排序](../BucketSort)一样,利用了“**桶**”的概念。基数排序不需要进行记录关键字间的比较,是一种**借助多关键字排序的思想对单逻辑关键字进行排序**的方法。比如数字100,它的个位、十位、百位就是不同的关键字。 + +那么,对于一组乱序的数字,基数排序的实现原理就是将整数按位数(关键字)切割成不同的数字,然后按每个位数分别比较。对于关键字的选择,有最高位优先法(MSD法)和最低位优先法(LSD法)两种方式。MSD 必须将序列先逐层分割成若干子序列,然后再对各子序列进行排序;而 LSD 进行排序时,不必分成子序列,对每个关键字都是整个序列参加排序。 + +### 算法步骤 + +以 LSD 法为例: +1. 将所有待比较数值(非负整数)统一为同样的数位长度,数位不足的数值前面补零 +2. 从最低位(个位)开始,依次进行一次排序 +3. 从最低位排序一直到最高位排序完成以后, 数列就变成一个有序序列 + +如果要支持负数参加排序,可以将序列中所有的值加上一个常数,使这些值都成为非负数,排好序后,所有的值再减去这个常数。 + +### 动图演示 + +![](radix-sort.gif) + +### 代码实现 + +#### C语言 +```c +// 基数,范围0~9 +#define RADIX 10 + +void radix_sort(int arr[], int n) { + // 获取最大值和最小值 + int max = arr[0], min = arr[0]; + int i, j, l; + for (i = 1; i < n; i++) { + if (max < arr[i]) max = arr[i]; + if (min > arr[i]) min = arr[i]; + } + // 假如序列中有负数,所有数加上一个常数,使序列中所有值变成正数 + if (min < 0) { + for (i = 0; i < n; i++) arr[i] -= min; + max -= min; + } + // 获取最大值位数 + int d = 0; + while (max > 0) { + max /= RADIX; + d ++; + } + int queue[RADIX][n]; + memset(queue, 0, sizeof(queue)); + int count[RADIX] = {0}; + for (i = 0; i < d; i++) { + // 分配数据 + for (j = 0; j < n; j++) { + int key = arr[j] % (int)pow(RADIX, i + 1) / (int)pow(RADIX, i); + queue[key][count[key]++] = arr[j]; + } + // 收集数据 + int c = 0; + for (j = 0; j < RADIX; j++) { + for (l = 0; l < count[j]; l++) { + arr[c++] = queue[j][l]; + queue[j][l] = 0; + } + count[j] = 0; + } + } + // 假如序列中有负数,收集排序结果时再减去前面加上的常数 + if (min < 0) { + for (i = 0; i < n; i++) arr[i] += min; + } +} +``` + +### 算法分析 + +基数排序是**稳定排序**,适用于关键字取值范围固定的排序。 + +#### 时间复杂度 + +基数排序可以看作是若干次“分配”和“收集”的过程。假设给定 n 个数,它的最高位数是 d,基数(也就是桶的个数)为 r,那么可以这样理解:共进行 d 趟排序,每趟排序都要对 n 个数据进行分配,再从 r 个桶中收集回来。所以算法的时间复杂度为 O(d(n+r)),在整数的排序中,r = 10,因此可以简化成 O(dn),是**线性阶**的排序。 + +#### 空间复杂度 + +占用额外内存,还需要 r 个桶,所以空间复杂度为 O(n+r)。 + +### 计数排序 & 桶排序 & 基数排序 + +这三种排序算法都利用了桶的概念,但对桶的使用方法上有明显差异: + +- 桶排序:每个桶存储一定范围的数值,适用于元素尽可能分布均匀的排序; +- 计数排序:每个桶只存储单一键值,适用于最大值和最小值尽可能接近的排序; +- 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key = arr[j] % (int)pow(RADIX, i + 1) / (int)pow(RADIX, i); + queue[key][count[key]++] = arr[j]; + } + // 收集数据 + int c = 0; + for (j = 0; j < RADIX; j++) { + for (l = 0; l < count[j]; l++) { + arr[c++] = queue[j][l]; + queue[j][l] = 0; + } + count[j] = 0; + } + } + // 假如序列中有负数,收集排序结果时再减去前面加上的常数 + if (min < 0) { + for (i = 0; i < n; i++) arr[i] += min; + } +} + +int main() { + int arr[] = {35, 10, 43, 3, 52, 21, 86, 19, 63, 60}; + int n = sizeof(arr) / sizeof(*arr); + radix_sort(arr, n); + printf("Sort result:\n"); + for (int i = 0; i < n; i++) + printf("%d ", arr[i]); + return 0; +}

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