快速排序的思想
任取一个元素作为枢轴，然后想办法把这个区间划分为两部分，小于等于枢轴的放左边，大于等于枢轴的放右边
然后递归处理左右区间，直到空或只剩一个
具体动画演示详见
数据结构合集 - 快速排序(算法过程, 效率分析, 稳定性分析)
Lomuto 分区方案（单边扫描法）
public static void quickSort(int[] nums){subSort(nums, 0, nums.length-1);
}private static void subSort(int[] nums, int low, int high){if(low < high){int pos = partition(nums, low, high);subSort(nums, low, pos-1);subSort(nums, pos+1, high);}
}private static int partition(int[] nums, int low, int high){int pivot = nums[high];int i = low-1;for(int j=low; j<high; j++){if(nums[j] <= pivot){i++;int tmp = nums[i];nums[i] = nums[j];nums[j] = tmp;}}// 将pivot放到正确的位置int temp = nums[i + 1];nums[i + 1] = nums[high];nums[high] = temp;return i+1;
}进一步优化上述代码
优化点：小数组时改用插入排序
当待排序数组较小的时候，快速排序的递归开销会比插入排序更大
public static void quickSort(int[] nums){subSort(nums, 0, nums.length-1);}private static void subSort(int[] nums, int low, int high) {if (low < high) {// 小数组改用插入排序（阈值可调整，通常 10~20）if (high - low < 10) {insertionSort(nums, low, high);return;}int pos = partition(nums, low, high);subSort(nums, low, pos - 1);subSort(nums, pos + 1, high);}}private static void insertionSort(int[] nums, int low, int high) {for (int i = low + 1; i <= high; i++) {int key = nums[i];int j = i - 1;while (j >= low && nums[j] > key) {nums[j + 1] = nums[j];j--;}nums[j + 1] = key;}}private static int partition(int[] nums, int low, int high){int pivot = nums[high];int i = low-1;for(int j=low; j<high; j++){if(nums[j] <= pivot){i++;int tmp = nums[i];nums[i] = nums[j];nums[j] = tmp;}}// 将pivot放到正确的位置int temp = nums[i + 1];nums[i + 1] = nums[high];nums[high] = temp;return i+1;}优化点：三数取中法选择pivot避免最坏情况
如果pivot选择最左或最右，在已排序或接近排序的数组上会导致最坏的情况O（）
public static void quickSort(int[] nums){subSort(nums, 0, nums.length-1);}private static void subSort(int[] nums, int low, int high) {if (low < high) {// 小数组改用插入排序（阈值可调整，通常 10~20）if (high - low < 10) {insertionSort(nums, low, high);return;}int pos = partition(nums, low, high);subSort(nums, low, pos - 1);subSort(nums, pos + 1, high);}}private static void insertionSort(int[] nums, int low, int high) {for (int i = low + 1; i <= high; i++) {int key = nums[i];int j = i - 1;while (j >= low && nums[j] > key) {nums[j + 1] = nums[j];j--;}nums[j + 1] = key;}}private static int partition(int[] nums, int low, int high){// 三数取中法选择 pivotint mid = low + (high - low) / 2;if (nums[low] > nums[mid]) swap(nums, low, mid);if (nums[low] > nums[high]) swap(nums, low, high);if (nums[mid] > nums[high]) swap(nums, mid, high);// 将中位数放到 nums[high]swap(nums, mid, high);int pivot = nums[high];int i = low-1;for(int j=low; j<high; j++){if(nums[j] <= pivot){i++;swap(nums, i, j);}}// 将pivot放到正确的位置swap(nums, i + 1, high);return i+1;}private static void swap(int[] nums, int i, int j) {int tmp = nums[i];nums[i] = nums[j];nums[j] = tmp;}优化点：尾递归优化避免栈溢出
public static void quickSort(int[] nums){subSort(nums, 0, nums.length-1);}private static void subSort(int[] nums, int low, int high) {// 小数组改用插入排序if (high - low < 10) {insertionSort(nums, low, high);return;}// 尾递归优化while (low < high) {int pos = partition(nums, low, high);if (pos - low < high - pos) {subSort(nums, low, pos - 1);low = pos + 1;} else {subSort(nums, pos + 1, high);high = pos - 1;}}}private static void insertionSort(int[] nums, int low, int high) {for (int i = low + 1; i <= high; i++) {int key = nums[i];int j = i - 1;while (j >= low && nums[j] > key) {nums[j + 1] = nums[j];j--;}nums[j + 1] = key;}}private static int partition(int[] nums, int low, int high){// 三数取中法选择 pivotint mid = low + (high - low) / 2;if (nums[low] > nums[mid]) swap(nums, low, mid);if (nums[low] > nums[high]) swap(nums, low, high);if (nums[mid] > nums[high]) swap(nums, mid, high);// 将中位数放到 nums[high]swap(nums, mid, high);int pivot = nums[high];int i = low-1;for(int j=low; j<high; j++){if(nums[j] <= pivot){i++;swap(nums, i, j);}}// 将pivot放到正确的位置swap(nums, i + 1, high);return i+1;}private static void swap(int[] nums, int i, int j) {int tmp = nums[i];nums[i] = nums[j];nums[j] = tmp;}优化点：双轴快排
java的Arrays.sort对小数组用插入排序，对大数组用双轴快排（比单轴更快）
双轴快排的基本思路：
- 选取两个 pivot：
-
pivot1 = nums[left]（较小的 pivot）pivot2 = nums[right]（较大的 pivot）- 确保
pivot1 ≤ pivot2（否则交换）
- 分区：
-
[left, i)：< pivot1[i, k)：≥ pivot1 && ≤ pivot2[k, j]：未处理区间(j, right]：> pivot2
- 递归处理三个子数组：
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[left, i-1]（小于pivot1的部分）[i, j]（介于pivot1和pivot2之间的部分）[j+1, right]（大于pivot2的部分）
public static void quickSort(int[] nums){dualPivotQuickSort(nums, 0, nums.length-1);}private static void dualPivotQuickSort(int[] nums, int left, int right) {if (left >= right) return;// 确保 pivot1 ≤ pivot2if (nums[left] > nums[right]) {swap(nums, left, right);}int pivot1 = nums[left];int pivot2 = nums[right];int i = left + 1; // [left+1, i) 存储 < pivot1 的元素int k = left + 1; // [i, k) 存储 ≥ pivot1 && ≤ pivot2 的元素int j = right - 1; // (j, right-1] 存储 > pivot2 的元素while (k <= j) {if (nums[k] < pivot1) {swap(nums, i, k);i++;k++;} else if (nums[k] <= pivot2) {k++;} else {swap(nums, k, j);j--;}}// 将 pivot1 和 pivot2 放到正确位置swap(nums, left, i - 1);swap(nums, right, j + 1);// 递归处理三个子数组dualPivotQuickSort(nums, left, i - 2); // < pivot1dualPivotQuickSort(nums, i, j); // pivot1 ≤ x ≤ pivot2dualPivotQuickSort(nums, j + 2, right); // > pivot2}private static void swap(int[] nums, int i, int j) {int tmp = nums[i];nums[i] = nums[j];nums[j] = tmp;}
