对于等于标定点的元素,也单独划分区间来保存,在下次循环的时候不考虑这部分区间,如果是元素相同的数组,就不会重复遍历这些元素了,这称为三路快速排序法

执行partition()方法前循环不变量:arr[left + 1, lt] < v,arr[lt + 1, i - 1] == v,arr[gt, right] > v

执行partition()方法后循环不变量:arr[left, lt - 1] < v,arr[lt, gt - 1] == v,arr[gt, right] > v

import java.util.Arrays;
import java.util.Random;

public class Algorithm {

    public static void main(String[] args) {

        Integer[] arr = {3, 2, 5, 1, 0, 0};
        QuickSort.sort3ways(arr);

        System.out.println(Arrays.toString(arr));
    }
}

class QuickSort {

    private QuickSort(){}

    public static<E extends Comparable<E>> void sort3ways(E[] arr){

        Random random = new Random();
        E temp = null;
        sort3ways(arr, 0, arr.length - 1, temp, random);
    }

    public static<E extends Comparable<E>> void sort3ways(E[] arr, int left, int right, E temp, Random random){

        if (left >= right){
            return;
        }

        int[] res = partition3ways(arr, left, right, temp, random);

        sort3ways(arr, left, res[0], temp, random);
        sort3ways(arr, res[1], right, temp, random);
    }

    /**
     * 循环不变量:arr[left + 1, lt] < v,arr[lt + 1, i - 1] == v,arr[gt, right] > v
     */
    public static<E extends Comparable<E>> int[] partition3ways(E[] arr, int left, int right, E temp, Random random){

        /**
         * 随机选择标定点,和arr[left]互换
         */
        int p = random.nextInt(right - left + 1) + left;

        swap(arr, p, left, temp);

        int i = left + 1;
        int lt = left;
        int gt = right + 1;

        /**
         * i == gt时,arr[gt]肯定大于arr[left],所以要结束循环
         */
        while (i < gt){

            /**
             * 当arr[i] < arr[left]时,将arr[i]和arr[lt + 1]互换,i继续向前
             */
            if (arr[i].compareTo(arr[left]) < 0){

                lt++;
                swap(arr, lt, i, temp);
                i++;
            }

            /**
             * 当arr[i] == arr[left]时,lt不动,i继续向前
             */
            else if (arr[i].compareTo(arr[left]) == 0){
                i++;
            }

            /**
             * 当arr[i] > arr[left]时,将arr[i]和arr[gt - 1]互换,i不动,因为换过来的元素还要进行判断
             */
            else if (arr[i].compareTo(arr[left]) > 0){
                gt--;
                swap(arr, gt, i, temp);
            }
        }

        /**
         * 遍历结束后,将arr[lt]和arr[left]互换,此时元素区间发生了改变
         * arr[left, lt - 1] < v,arr[lt, gt - 1] == v,arr[gt, right] > v
         * 其中等于arr[left]的区间不用再返回,只需返回两侧区间的索引
         */
        swap(arr, lt, left, temp);

        int[] res = {lt - 1, gt};

        return res;
    }

    public static<E extends Comparable<E>> void swap(E[] arr, int i, int j, E temp){

        temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }
}

三路快速排序法的复杂度分析

在元素都相同的数组中,三路快速排序法的时间复杂度为O(n)

双路和三路快速排序法性能比较

import java.util.Arrays;
import java.util.Random;

public class Algorithm {

    public static void main(String[] args) {

        Integer[] testScale = {10000, 100000};

        for (Integer n : testScale){

            Integer[] randomArr = ArrayGenerator.generatorRandomArray(n, 1);
            Integer[] sortedArr = ArrayGenerator.generatorSortedArray(n, n);

            Integer[] arr1 = Arrays.copyOf(randomArr, randomArr.length);
            Integer[] arr3 = Arrays.copyOf(randomArr, randomArr.length);

            Integer[] arr2 = Arrays.copyOf(sortedArr, sortedArr.length);
            Integer[] arr4 = Arrays.copyOf(sortedArr, sortedArr.length);

            System.out.println("测试随机数组排序性能");
            System.out.println();

            Verify.testTime("QuickSort2Ways", arr1);
            Verify.testTime("QuickSort3Ways", arr3);

            System.out.println();

            System.out.println("测试有序数组排序性能");
            System.out.println();

            Verify.testTime("QuickSort2Ways", arr2);
            Verify.testTime("QuickSort3Ways", arr4);

            System.out.println();
        }
    }
}

class QuickSort {

    private QuickSort() {
    }

    public static <E extends Comparable<E>> void sort2ways(E[] arr) {

        Random random = new Random();
        E temp = null;
        sort2ways(arr, 0, arr.length - 1, temp, random);
    }

    public static <E extends Comparable<E>> void sort2ways(E[] arr, int left, int right, E temp, Random random) {

        if (left >= right) {
            return;
        }

        int p = partition2ways(arr, left, right, temp, random);

        sort2ways(arr, left, p - 1, temp, random);
        sort2ways(arr, p + 1, right, temp, random);
    }

    public static <E extends Comparable<E>> int partition2ways(E[] arr, int left, int right, E temp, Random random) {

        int p = random.nextInt(right - left + 1) + left;

        temp = arr[p];
        arr[p] = arr[left];
        arr[left] = temp;

        int i = left + 1;
        int j = right;

        while (i <= j) {

            if (arr[i].compareTo(arr[left]) < 0) {
                i++;
            } else if (arr[j].compareTo(arr[left]) > 0) {
                j--;
            } else {
                temp = arr[i];
                arr[i] = arr[j];
                arr[j] = temp;
                i++;
                j--;
            }
        }

        temp = arr[j];
        arr[j] = arr[left];
        arr[left] = temp;

        return j;
    }

    public static<E extends Comparable<E>> void sort3ways(E[] arr){

        Random random = new Random();
        E temp = null;
        sort3ways(arr, 0, arr.length - 1, temp, random);
    }

    public static<E extends Comparable<E>> void sort3ways(E[] arr, int left, int right, E temp, Random random){

        if (left >= right){
            return;
        }

        int[] res = partition3ways(arr, left, right, temp, random);

        sort3ways(arr, left, res[0], temp, random);
        sort3ways(arr, res[1], right, temp, random);
    }

    public static<E extends Comparable<E>> int[] partition3ways(E[] arr, int left, int right, E temp, Random random){

        int p = random.nextInt(right - left + 1) + left;

        swap(arr, p, left, temp);

        int i = left + 1;
        int lt = left;
        int gt = right + 1;

        while (i < gt){

            if (arr[i].compareTo(arr[left]) < 0){

                lt++;
                swap(arr, lt, i, temp);
                i++;
            }

            else if (arr[i].compareTo(arr[left]) == 0){
                i++;
            }

            else if (arr[i].compareTo(arr[left]) > 0){
                gt--;
                swap(arr, gt, i, temp);
            }
        }

        swap(arr, lt, left, temp);

        int[] res = {lt - 1, gt};

        return res;
    }

    public static<E extends Comparable<E>> void swap(E[] arr, int i, int j, E temp){

        temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }
}

class ArrayGenerator {

    private ArrayGenerator (){}

    public static Integer[] generatorRandomArray (Integer n, Integer maxBound){

        Integer[] arr = new Integer[n];

        Random random = new Random();

        for (int i = 0; i < n; i++) {

            arr[i] = random.nextInt(maxBound);
        }

        return arr;
    }

    public static Integer[] generatorSortedArray (Integer n, Integer maxBound){

        Integer[] arr = new Integer[n];

        for (int i = 0; i < n; i++) {

            arr[i] = i;
        }

        return arr;
    }
}

class Verify {

    private Verify (){}

    public static<E extends Comparable<E>> boolean isSorted(E[] arr){

        for (int i = 0; i < arr.length - 1; i++) {
            if (arr[i].compareTo(arr[i + 1]) > 0) {
                return false;
            }
        }

        return true;
    }

    public static<E extends Comparable<E>> void testTime(String AlgorithmName, E[] arr) {

        long startTime = System.nanoTime();

        if (AlgorithmName.equals("QuickSort2Ways")) {
            QuickSort.sort2ways(arr);
        }

        if (AlgorithmName.equals("QuickSort3Ways")) {
            QuickSort.sort3ways(arr);
        }

        long endTime = System.nanoTime();

        if (!Verify.isSorted(arr)){
            throw new RuntimeException(AlgorithmName + "算法排序失败!");
        }

        System.out.println(String.format("%s算法,测试用例为%d,执行时间:%f秒", AlgorithmName, arr.length, (endTime - startTime) / 1000000000.0));
    }
}