归并排序法03：自底向上实现归并排序法

之前的归并排序称为自顶向下，将一个大的数组一步步的拆分为小的区间，分别进行排序，最后再合并

也可以将数组的元素拆分为n个数组，第一次比较只有一个元素的两个分区间，第二次再比较两元素的分区间，第三次比较四个元素...

这种从小区间组合成大数组的方式，称为自底向上排序（Bottom to Up，这个过程不会使用到递归）

import java.util.Arrays;

public class Algorithm {

    public static void main(String[] args) {

        Integer[] arr = {6,6,3,2,4,7};

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

class MergeSort {

    private MergeSort(){}

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

        E[] temp = Arrays.copyOf(arr, arr.length);

        /**
         * 遍历要合并的子区间长度，每次长度翻倍
         */
        for (int size = 1; size < arr.length; size *= 2) {

            /**
             * 遍历两个子区间的起始位置，每次相比上次偏移了2倍长度
             * 两个子区间为arr[i, i + size - 1]和arr[i + size, i + size + size - 1]
             * 注意：数组元素个数不一定为2的整数幂，即最后得到的两个分区间元素个数可能不同，因此i + size + size - 1最终是有可能越界的，需要和数组长度二者来取最小值
             */
            for (int i = 0; i + size < arr.length; i = i + 2 * size) {
                if (arr[i + size - 1].compareTo(arr[i + size]) > 0) {
                    mergeBU(arr, i, i + size - 1, Math.min(arr.length - 1, i + 2 * size - 1), temp);
                }
            }
        }
    }

    public static<E extends Comparable<E>> void mergeBU(E[] arr, int left, int mid, int right, E[] temp) {

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

        System.arraycopy(arr, left, temp, left, right - left + 1);

        for (int n = left; n < right + 1; n++) {

            if (i == mid + 1){
                arr[n] = temp[j];
                j++;
            }
            else if (j == right + 1) {
                arr[n] = temp[i];
                i++;
            }
            else if (temp[i].compareTo(temp[j]) <= 0) {
                arr[n] = temp[i];
                i++;
            }
            else{
                arr[n] = temp[j];
                j++;
            }
        }
    }
}

使用插入排序法优化

在子区间长度小于16的时候，用插入排序法来实现

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

public class Algorithm {

    public static void main(String[] args) {

        Integer[] testScale = {10000, 500000};

        for (Integer n : testScale) {

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

            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("sortBU", arr1);
            Verify.testTime("sortBU1", arr3);

            System.out.println();

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

            Verify.testTime("sortBU", arr2);
            Verify.testTime("sortBU1", arr4);

            System.out.println();
        }
    }
}

class MergeSort {

    private MergeSort() {
    }

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

        E[] temp = Arrays.copyOf(arr, arr.length);

        for (int size = 1; size < arr.length; size *= 2) {

            for (int i = 0; i + size < arr.length; i = i + 2 * size) {
                if (arr[i + size - 1].compareTo(arr[i + size]) > 0) {
                    mergeBU(arr, i, i + size - 1, Math.min(arr.length - 1, i + 2 * size - 1), temp);
                }
            }
        }
    }

    /**
     * 将数组分为长度为16的子区间，用插入排序法实现排序
     * 然后从长度16开始，用归并排序法进行合并
     */
    public static <E extends Comparable<E>> void sortBU1(E[] arr) {

        E[] temp = Arrays.copyOf(arr, arr.length);

        for (int i = 0; i < arr.length; i += 16){
            InsertionSort.sort(arr, i, Math.min(i + 15, arr.length - 1));
        }

        for (int size = 16; size < arr.length; size *= 2) {

            for (int i = 0; i + size < arr.length; i = i + 2 * size) {
                if (arr[i + size - 1].compareTo(arr[i + size]) > 0) {
                    mergeBU(arr, i, i + size - 1, Math.min(arr.length - 1, i + 2 * size - 1), temp);
                }
            }
        }
    }

    public static <E extends Comparable<E>> void mergeBU(E[] arr, int left, int mid, int right, E[] temp) {

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

        System.arraycopy(arr, left, temp, left, right - left + 1);

        for (int n = left; n < right + 1; n++) {

            if (i == mid + 1) {
                arr[n] = temp[j];
                j++;
            } else if (j == right + 1) {
                arr[n] = temp[i];
                i++;
            } else if (temp[i].compareTo(temp[j]) <= 0) {
                arr[n] = temp[i];
                i++;
            } else {
                arr[n] = temp[j];
                j++;
            }
        }
    }
}

class InsertionSort {

    private InsertionSort() {}

    public static <E extends Comparable> void sort(E[] arr, int left, int right) {

        for (int i = left + 1; i < right + 1; i++) {

            E tem = arr[i];
            int j;

            for (j = i; j > left && tem.compareTo(arr[j - 1]) < 0; j--) {

                arr[j] = arr[j - 1];
            }

            arr[j] = tem;
        }
    }
}

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("sortBU")) {
            MergeSort.sortBU(arr);
        }

        if (AlgorithmName.equals("sortBU1")) {
            MergeSort.sortBU1(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));
    }
}

自顶向下和自底向上性能比较

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

public class Algorithm {

    public static void main(String[] args) {

        Integer[] testScale = {10000, 500000};

        for (Integer n : testScale) {

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

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

            Integer[] arr1 = Arrays.copyOf(randomArr, randomArr.length);
            Integer[] arr3 = Arrays.copyOf(randomArr, randomArr.length);
            Integer[] arr5 = Arrays.copyOf(randomArr, randomArr.length);

            Integer[] arr2 = Arrays.copyOf(sortedArr, sortedArr.length);
            Integer[] arr4 = Arrays.copyOf(sortedArr, sortedArr.length);
            Integer[] arr6 = Arrays.copyOf(sortedArr, sortedArr.length);

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

            Verify.testTime("sortOptimised1", arr1);
            Verify.testTime("sortOptimised2", arr3);
            Verify.testTime("sortBU", arr5);

            System.out.println();

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

            Verify.testTime("sortOptimised1", arr2);
            Verify.testTime("sortOptimised2", arr4);
            Verify.testTime("sortBU", arr6);

            System.out.println();
        }
    }
}

class MergeSort {

    private MergeSort(){}

    /**
     * 优化一：有序数组优化，合并前先判断一下是否需要合并
     */
    public static<E extends Comparable<E>> void sortOptimised1(E[] arr){

        sortOptimised1(arr, 0, arr.length - 1);
    }

    private static<E extends Comparable<E>> void sortOptimised1(E[] arr, int left, int right){

        if (left >= right){
            return;
        }

        int mid = left + (right - left) / 2;

        sortOptimised1(arr, left, mid);
        sortOptimised1(arr, mid + 1, right);


        if (arr[mid].compareTo(arr[mid + 1]) > 0) {
            merge(arr, left, mid, right);
        }
    }

    /**
     * 优化三：内存操作优化
     */
    public static<E extends Comparable<E>> void sortOptimised2(E[] arr){

        /**
         * 提前将arr数组保存一个副本，这样就不用每次调用merge()方法，都重新开辟空间新建一个数组了
         */
        E[] temp = Arrays.copyOf(arr, arr.length);

        sortOptimised2(arr, 0, arr.length - 1, temp);
    }

    private static<E extends Comparable<E>> void sortOptimised2(E[] arr, int left, int right, E[] temp){

        if (left >= right){
            return;
        }

        int mid = left + (right - left) / 2;

        sortOptimised2(arr, left, mid, temp);
        sortOptimised2(arr, mid + 1, right, temp);

        if (arr[mid].compareTo(arr[mid + 1]) > 0) {
            mergeOptimised(arr, left, mid, right, temp);
        }
    }

    /**
     * 自底向上归并排序法
     */
    public static<E extends Comparable<E>> void sortBU(E[] arr){

        E[] temp = Arrays.copyOf(arr, arr.length);

        for (int size = 1; size < arr.length; size *= 2) {

            for (int i = 0; i + size < arr.length; i = i + 2 * size) {
                if (arr[i + size - 1].compareTo(arr[i + size]) > 0) {
                    mergeBU(arr, i, i + size - 1, Math.min(arr.length - 1, i + 2 * size - 1), temp);
                }
            }
        }
    }

    public static<E extends Comparable<E>> void mergeBU(E[] arr, int left, int mid, int right, E[] temp) {

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

        System.arraycopy(arr, left, temp, left, right - left + 1);

        for (int n = left; n < right + 1; n++) {

            if (i == mid + 1){
                arr[n] = temp[j];
                j++;
            }
            else if (j == right + 1) {
                arr[n] = temp[i];
                i++;
            }
            else if (temp[i].compareTo(temp[j]) <= 0) {
                arr[n] = temp[i];
                i++;
            }
            else{
                arr[n] = temp[j];
                j++;
            }
        }
    }

    public static<E extends Comparable<E>> void merge(E[] arr, int left, int mid, int right) {

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

        E[] tem = Arrays.copyOfRange(arr, left, right + 1);

        for (int n = left; n < right + 1; n++) {

            if (i == mid + 1){
                arr[n] = tem[j - left];
                j++;
            }
            else if (j == right + 1) {
                arr[n] = tem[i - left];
                i++;
            }
            else if (tem[i - left].compareTo(tem[j - left]) <= 0) {
                arr[n] = tem[i - left];
                i++;
            }
            else{
                arr[n] = tem[j - left];
                j++;
            }
        }
    }

    /**
     * 优化merge()方法，复用temp数组，节省空间
     */
    public static<E extends Comparable<E>> void mergeOptimised(E[] arr, int left, int mid, int right, E[] temp) {

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

        /**
         * System.arraycopy()方法将传过来的排好序的两个分数组在相同位置赋值给副本数组temp，因此索引范围一致没有偏移
         */
        System.arraycopy(arr, left, temp, left, right - left + 1);

        for (int n = left; n < right + 1; n++) {

            if (i == mid + 1){
                arr[n] = temp[j];
                j++;
            }
            else if (j == right + 1) {
                arr[n] = temp[i];
                i++;
            }
            else if (temp[i].compareTo(temp[j]) <= 0) {
                arr[n] = temp[i];
                i++;
            }
            else{
                arr[n] = temp[j];
                j++;
            }
        }
    }
}

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("sortOptimised1")) {
            MergeSort.sortOptimised1(arr);
        }

        if (AlgorithmName.equals("sortOptimised2")) {
            MergeSort.sortOptimised2(arr);
        }

        if (AlgorithmName.equals("sortBU")) {
            MergeSort.sortBU(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));
    }
}