以下是几种 C++ 实现且时间复杂度在 $O(nlogn)$ 及以内的算法：
### 归并排序
归并排序是一种稳定的排序算法，其时间复杂度为 $O(nlogn)$，空间复杂度是 $O(n)$，主要占用空间的是排序前创建的长度为 $n$ 的辅助数组。代码如下：
```cpp
#include <iostream>
#include <vector>
void merge(std::vector<int>& arr, int left, int mid, int right) {
int n1 = mid - left + 1;
int n2 = right - mid;
std::vector<int> L(n1), R(n2);
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];
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++;
}
while (i < n1) {
arr[k] = L[i];
i++;
k++;
}
while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
void mergeSort(std::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() {
std::vector<int> arr = {12, 11, 13, 5, 6, 7};
int n = arr.size();
mergeSort(arr, 0, n - 1);
for (int i = 0; i < n; i++)
std::cout << arr[i] << " ";
std::cout << std::endl;
return 0;
}
```
归并排序在拆分数组的过程中，会将数组拆分 $logn$ 次，每层执行的比较次数都约等于 $n$ 次，所以时间复杂度是 $O(nlogn)$ [^4]。
### 快速排序
快速排序是一种不稳定的排序算法，平均时间复杂度为 $O(nlogn)$，最坏情况下时间复杂度为 $O(n^2)$。代码如下：
```cpp
#include <iostream>
#include <vector>
int partition(std::vector<int>& arr, int low, int high) {
int pivot = arr[high];
int i = (low - 1);
for (int j = low; j <= high - 1; j++) {
if (arr[j] < pivot) {
i++;
std::swap(arr[i], arr[j]);
}
}
std::swap(arr[i + 1], arr[high]);
return (i + 1);
}
void quickSort(std::vector<int>& arr, int low, int high) {
if (low < high) {
int pi = partition(arr, low, high);
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
int main() {
std::vector<int> arr = {10, 7, 8, 9, 1, 5};
int n = arr.size();
quickSort(arr, 0, n - 1);
for (int i = 0; i < n; i++)
std::cout << arr[i] << " ";
std::cout << std::endl;
return 0;
}
```
快速排序的平均时间复杂度为 $O(nlogn)$，适合大规模的数据排序 [^2]。
### 堆排序
堆排序也是一种时间复杂度为 $O(nlogn)$ 的排序算法。代码如下：
```cpp
#include <iostream>
#include <vector>
void heapify(std::vector<int>& arr, int n, int i) {
int largest = i;
int l = 2 * i + 1;
int r = 2 * i + 2;
if (l < n && arr[l] > arr[largest])
largest = l;
if (r < n && arr[r] > arr[largest])
largest = r;
if (largest != i) {
std::swap(arr[i], arr[largest]);
heapify(arr, n, largest);
}
}
void heapSort(std::vector<int>& arr) {
int n = arr.size();
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
for (int i = n - 1; i > 0; i--) {
std::swap(arr[0], arr[i]);
heapify(arr, i, 0);
}
}
int main() {
std::vector<int> arr = {12, 11, 13, 5, 6, 7};
int n = arr.size();
heapSort(arr);
for (int i = 0; i < n; i++)
std::cout << arr[i] << " ";
std::cout << std::endl;
return 0;
}
```
### 二分查找
二分查找的时间复杂度为 $O(logn)$，前提是数组是有序的。代码如下：
```cpp
#include <iostream>
#include <vector>
int binarySearch(const std::vector<int>& arr, int target) {
int left = 0, right = arr.size() - 1;
while (left <= right) {
int mid = left + (right - left) / 2;
if (arr[mid] == target) {
return mid;
} else if (arr[mid] < target) {
left = mid + 1;
} else {
right = mid - 1;
}
}
return -1;
}
int main() {
std::vector<int> arr = {1, 2, 3, 4, 5, 6, 7};
int target = 5;
int result = binarySearch(arr, target);
if (result != -1) {
std::cout << "Element found at index " << result << std::endl;
} else {
std::cout << "Element not found" << std::endl;
}
return 0;
}
```
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