排序3-归并排序

基本思想

归并排序（Merge Sort）完全遵循上述分治法三个步骤：
1、分解：将要排序的n个元素的序列分解成两个具有n/2个元素的子序列；
2、解决：使用归并排序分别递归地排序两个子序列；
3、合并：合并两个已排序的子序列，产生原问题的解。
所以说归并排序一种分治算法的典型应用。
归并排序过程动态演示

代码实现

void merge_array(int arr[], int tmp[], int Left, int mid, int Right)
{
    assert(arr && Left >= 0 && Left <= mid && mid <= Right);
    int i = Left;
    int j = mid + 1;
    int index = 0;

    while(i <= mid && j <= Right)
    {
        if(arr[i] <= arr[j])
            tmp[index++] = arr[i++];
        else
            tmp[index++] = arr[j++];
    }

    while(i <= mid) //拷贝剩下的左半部分
        tmp[index++] = arr[i++];

    while(j <= Right)  //拷贝剩下的右半部分
        tmp[index++] = arr[j++];

    memcpy(arr + Left, tmp, (Right - Left + 1) * sizeof(int)); 
}
void merge_sort(int arr[], int tmp[], int Left, int Right)
{
    assert(arr && Left >= 0);
    int mid;

    if(Left < Right)
    {
        mid = (Left + Right) / 2;
        merge_sort(arr, tmp, Left, mid);
        merge_sort(arr, tmp, mid + 1, Right);
        merge_array(arr, tmp, Left, mid, Right);
    }

}

void Merge_Sort(int arr[], int len)
{
    assert(arr && len);
    int *tmp_arr;
    //tmp_arr = (int *)malloc(len * sizeof(int));
    tmp_arr = new int[len];

    merge_sort(arr, tmp_arr, 0, len - 1);

    delete[] tmp_arr;

}

测试程序

#include <iostream>
#include <cstring>
#include <ctime>
#include <cmath>
using namespace std;


#define ArraySize 100000


void swap(int *x, int *y)
{
    int temp;
    temp = *x;
    *x   = *y;
    *y   = temp;
}

void Bubble_sort(int arr[], int len)
{
    for(int i = 0; i < len; i ++)
    {
        for(int j = i + 1; j < len; j ++)
            if(arr[i] > arr[j])
                swap(arr[i], arr[j]);
    }
}

void Bubble_sort1(int arr[], int len)
{
    for(int i = 0; i < len; i ++)
    {
        for(int j = len - 1; j >= i; j --)
        {
            if(arr[i] > arr[j])
                swap(arr[i], arr[j]);
        }
    }
}

void Bubble_sort2(int arr[], int len)
{
    bool flag = true;

    while(flag)
    {
        flag = false;
        for(int i = 0; i < len; i ++)
            for(int j = len - 1; j >= i; j --)
                if(arr[i] > arr[j])
                    swap(arr[i], arr[j]);
    }
}

void Slect_sort(int arr[], int len)
{
    for(int i = 0; i < len; i ++)
    {
        int min_index = i ;
        for(int j = i + 1; j < len; j ++)
        {
            if(arr[min_index] > arr[j])
                min_index = j;
        }

        if(i != min_index)
            swap(arr[i],arr[min_index]);
    }
}

void Insert_sort(int arr[], int len)
{

    for(int i= 1; i < len; i ++)
    {   
        int key = arr[i];
        int j = i;
        while(j && arr[j - 1] > key)
        {
            arr[j] = arr[j - 1];
            j --;
        }

        arr[j] = key;
    }
}

void Shell_sort(int arr[], int len)
{
    int increment = len / 2;

    while(increment)
    {
        for(int i = increment; i < len; i ++)
        {
            int key = arr[i];
            /*int j ;
            for(j = i; j >= increment; j -= increment)
            {
                if(arr[j-increment] > key )
                    arr[j] = arr[j-increment];
                else 
                    break;

            }*/

            int j = i;
            while(j >= increment && arr[j-increment] > key)
            {
                arr[j] = arr[j-increment];
                j -= increment;
            }

            arr[j] = key;
        }

        increment /= 2;
    }
}

void Shell_sort1(int arr[], int len)
{
    int increment = 0;
    for(increment = len/2; increment > 0; increment /=2)
    {
        for(int i = increment; i < len; i++)
        {
            int key = arr[i];
            int j = 0;
            for(j = i; j >= increment; j -=increment)
            {
                if(arr[j-increment] > key)
                    arr[j] = arr[j-increment];
                else 
                    break;
            }

            arr[j] = key;
        }
    }
}

void Shell_sort2(int arr[], int len)
{
    int index = log( 2*len + 1) / log(3.0);

    //cout << index << endl;

    int increment = ( pow(3.0, index) - 1 ) / 2;

    //cout << increment << endl;

    while(increment)
    {
        for(int i = increment; i < len; i ++)
        {
            int key = arr[i];
            /*int j ;
            for(j = i; j >= increment; j -= increment)
            {
                if(arr[j-increment] > key )
                    arr[j] = arr[j-increment];
                else 
                    break;

            }*/

            int j = i;
            while(j >= increment && arr[j-increment] > key)
            {
                arr[j] = arr[j-increment];
                j -= increment;
            }

            arr[j] = key;
        }
        index -= 1;
        increment = ( pow(3.0, index) - 1 ) / 2;
    }
}

void Heap_adjust(int arr[], int index, int len)
{


    while(true)
    {
        int iMax = index;
        int iLeft = 2 * index + 1;
        int iRight = 2 * index + 2;

        if(iLeft < len && arr[index] < arr[iLeft])
            iMax = iLeft;
        if(iRight < len && arr[index] < arr[iRight])
            iMax = iRight;
        if(iMax != index)
        {
            swap(arr[index], arr[iMax]);
            index = iMax;
        }
        else 
            break;
    }



}

void Heap_adjust2(int arr[], int index, int len)
{
    int iMax = index;
    int iLeft = 2 * index + 1;
    int iRight = 2 * index + 2;

    if(iLeft < len && arr[index] < arr[iLeft])
        iMax = iLeft;

    if(iRight < len && arr[index] < arr[iRight])
        iMax = iRight;

    if(iMax != index)
    {
        swap(arr[index], arr[iMax]);
        Heap_adjust2(arr, iMax, len);
    }
}

void Build_maxheap(int arr[], int len)
{
    for(int i = len / 2; i >= 0; i --)
    {
        Heap_adjust(arr, i , len);
    }
}

void Heap_Sort(int arr[], int len)
{
    Build_maxheap(arr, len);

    for(int i = len - 1; i > 0; i --)
    {
        swap(arr[0], arr[i]);
        Heap_adjust(arr, 0, i);
    }
}

void Print_array(int arr[], int len)
{
    for(int i = 0; i < len; i++)
    {
        cout << arr[i] << " ";
    }
    cout << endl;

}

void merge_array(int arr[], int tmp[], int Left, int mid, int Right)
{
    assert(arr && Left >= 0 && Left <= mid && mid <= Right);
    int i = Left;
    int j = mid + 1;
    int index = 0;

    while(i <= mid && j <= Right)
    {
        if(arr[i] <= arr[j])
            tmp[index++] = arr[i++];
        else
            tmp[index++] = arr[j++];
    }

    while(i <= mid)
        tmp[index++] = arr[i++];

    while(j <= Right)
        tmp[index++] = arr[j++];

    memcpy(arr + Left, tmp, (Right - Left + 1) * sizeof(int)); 
}
void merge_sort(int arr[], int tmp[], int Left, int Right)
{
    assert(arr && Left >= 0);
    int mid;

    if(Left < Right)
    {
        mid = (Left + Right) / 2;
        merge_sort(arr, tmp, Left, mid);
        merge_sort(arr, tmp, mid + 1, Right);
        merge_array(arr, tmp, Left, mid, Right);
    }

}

void Merge_Sort(int arr[], int len)
{
    assert(arr && len);
    int *tmp_arr;
    //tmp_arr = (int *)malloc(len * sizeof(int));
    tmp_arr = new int[len];

    merge_sort(arr, tmp_arr, 0, len - 1);

    delete[] tmp_arr;

}
int main(int argc, char const *argv[])
{
    /* code */
    int Array[ArraySize];
    int Array1[ArraySize];
    int Array2[ArraySize];

    time_t begin , end;

    srand(time(NULL));



    for(int i = 0; i < ArraySize; i ++)
    {
        Array[i] = rand()%ArraySize;
        //cout << Array[i] << " ";
    }

    memcpy(Array1, Array, ArraySize * sizeof(Array1[0]));
    memcpy(Array2, Array, ArraySize * sizeof(Array2[0]));

//  Print_array(Array, ArraySize);

/*  begin = clock();
    Bubble_sort2(Array, ArraySize);
    end = clock();
    cout << "Bubble_sort runtime:   " << double(end - begin) / CLOCKS_PER_SEC << "s" << endl;

    begin = clock();
    Slect_sort(Array1, ArraySize);
    end = clock();
    cout << "Slect_sort runtime:   " << double(end - begin) / CLOCKS_PER_SEC << "s" << endl;

    begin = clock();
    Insert_sort(Array2, ArraySize);
    end = clock();
    cout << "Insert_sort runtime:   " << double(end - begin) / CLOCKS_PER_SEC << "s" << endl;*/ 

    begin = clock();
    Shell_sort2(Array, ArraySize);
    end = clock();
    cout << "Shell_sort2 runtime:   " << double(end - begin) / CLOCKS_PER_SEC << "s" << endl;

    begin = clock();
    Heap_Sort(Array1, ArraySize);
    end = clock();
    cout << "Heap_Sort runtime:   " << double(end - begin) / CLOCKS_PER_SEC << "s" << endl;

    begin = clock();
    Merge_Sort(Array2, ArraySize);
    end = clock();
    cout << "Merge_Sort runtime:   " << double(end - begin) / CLOCKS_PER_SEC << "s" << endl;

    //Print_array(Array2, ArraySize);
    return 0;
}

运行结果如下：

Shell_sort2 runtime:   0.026s
Heap_Sort runtime:   0.004s
Merge_Sort runtime:   0.014s

时间和空间复杂度

时间复杂度是$O(N\log{N})$，空间复制度为$O(N)$（归并排序的最大缺陷）。归并排序在数据量比较大的时候也有较为出色的表现（效率上），但是，其空间复杂度O(n)使得在数据量特别大的时候（例如，1千万数据）几乎不可接受。而且，考虑到有的机器内存本身就比较小。总结来说，归并排序是一种占用内存，但却效率高且稳定的算法。