排序算法

排序算法简介

排序算法将元素按特定顺序排列，通常是升序或降序。理解排序是计算机科学和算法设计的基础。

冒泡排序

冒泡排序反复遍历列表，比较相邻元素，如果顺序错误就交换它们。时间复杂度：O(n²)

冒泡排序实现

#include <iostream>
#include <vector>
using namespace std;

void bubbleSort(vector<int>& arr) {
    int n = arr.size();
    
    for (int i = 0; i < n - 1; i++) {
        bool swapped = false;
        
        // Last i elements are already sorted
        for (int j = 0; j < n - i - 1; j++) {
            if (arr[j] > arr[j + 1]) {
                swap(arr[j], arr[j + 1]);
                swapped = true;
            }
        }
        
        // If no swapping occurred, array is sorted
        if (!swapped) break;
    }
}

int main() {
    vector<int> arr = {64, 34, 25, 12, 22, 11, 90};
    
    cout << "Original array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    bubbleSort(arr);
    
    cout << "Sorted array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    return 0;
}

选择排序

选择排序找到最小元素并将其放在开头，然后找到第二小的元素，依此类推。时间复杂度：O(n²)

选择排序实现

#include <iostream>
#include <vector>
using namespace std;

void selectionSort(vector<int>& arr) {
    int n = arr.size();
    
    for (int i = 0; i < n - 1; i++) {
        int minIndex = i;
        
        // Find minimum element in remaining array
        for (int j = i + 1; j < n; j++) {
            if (arr[j] < arr[minIndex]) {
                minIndex = j;
            }
        }
        
        // Swap minimum element with first element
        if (minIndex != i) {
            swap(arr[i], arr[minIndex]);
        }
    }
}

int main() {
    vector<int> arr = {64, 34, 25, 12, 22, 11, 90};
    
    cout << "Original array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    selectionSort(arr);
    
    cout << "Sorted array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    return 0;
}

快速排序

快速排序使用分治法选择一个基准元素并对数组进行分区。平均时间复杂度：O(n log n)，最坏情况：O(n²)

快速排序实现

#include <iostream>
#include <vector>
using namespace std;

int partition(vector<int>& arr, int low, int high) {
    int pivot = arr[high];  // Choose last element as pivot
    int i = low - 1;        // Index of smaller element
    
    for (int j = low; j < high; j++) {
        if (arr[j] <= pivot) {
            i++;
            swap(arr[i], arr[j]);
        }
    }
    
    swap(arr[i + 1], arr[high]);
    return i + 1;
}

void quickSort(vector<int>& arr, int low, int high) {
    if (low < high) {
        int pi = partition(arr, low, high);
        
        // Recursively sort elements before and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

int main() {
    vector<int> arr = {64, 34, 25, 12, 22, 11, 90};
    
    cout << "Original array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    quickSort(arr, 0, arr.size() - 1);
    
    cout << "Sorted array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    return 0;
}

归并排序

归并排序将数组分成两半，递归地对它们排序，然后合并已排序的两半。时间复杂度：O(n log n) - 稳定且可预测。

归并排序实现

#include <iostream>
#include <vector>
using namespace std;

void merge(vector<int>& arr, int left, int mid, int right) {
    int n1 = mid - left + 1;
    int n2 = right - mid;
    
    // Create temporary arrays
    vector<int> L(n1), R(n2);
    
    // Copy data to temporary arrays
    for (int i = 0; i < n1; i++)
        L[i] = arr[left + i];
    for (int j = 0; j < n2; j++)
        R[j] = arr[mid + 1 + j];
    
    // Merge the temporary arrays back
    int i = 0, j = 0, k = left;
    
    while (i < n1 && j < n2) {
        if (L[i] <= R[j]) {
            arr[k] = L[i];
            i++;
        } else {
            arr[k] = R[j];
            j++;
        }
        k++;
    }
    
    // Copy remaining elements
    while (i < n1) {
        arr[k] = L[i];
        i++;
        k++;
    }
    
    while (j < n2) {
        arr[k] = R[j];
        j++;
        k++;
    }
}

void mergeSort(vector<int>& arr, int left, int right) {
    if (left < right) {
        int mid = left + (right - left) / 2;
        
        mergeSort(arr, left, mid);
        mergeSort(arr, mid + 1, right);
        merge(arr, left, mid, right);
    }
}

int main() {
    vector<int> arr = {64, 34, 25, 12, 22, 11, 90};
    
    cout << "Original array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    mergeSort(arr, 0, arr.size() - 1);
    
    cout << "Sorted array: ";
    for (int x : arr) cout << x << " ";
    cout << endl;
    
    return 0;
}

复杂度比较

算法	最佳情况	平均情况	最坏情况	空间复杂度
冒泡排序	O(n)	O(n²)	O(n²)	O(1)
选择排序	O(n²)	O(n²)	O(n²)	O(1)
快速排序	O(n log n)	O(n log n)	O(n²)	O(log n)
归并排序	O(n log n)	O(n log n)	O(n log n)	O(n)