算法基礎——歸併排序

1. 思想

分治法：

1. 選擇中間點
2. 遞歸排序左邊、右邊
3. 歸併

2. 模板

#include <iostream>

using namespace std;

const int N = 1e6+10;
int arr[N],tmp[N];

void mergeSort(int *arr, int l, int r)
{
    if(l >= r)  return ;
    
    int mid = (l+r)>>1;					//1. 選擇中間點
    mergeSort(arr, l, mid);				//2. 遞歸排序左邊、右邊 
    mergeSort(arr, mid+1, r);
    
    int k = 0, i = l, j = mid + 1;
    while(i<=mid && j<=r)				//3. 歸併
    {
        if(arr[i] < arr[j])
            tmp[k++] = arr[i++];
        else
            tmp[k++] = arr[j++];
    }
    
    while(i <= mid) tmp[k++] = arr[i++];
    while(j <= r)   tmp[k++] = arr[j++];
    
    for(int i=0, j=l; j<=r; i++,j++)  arr[j] = tmp[i];
}

int main()
{
    int n;
    scanf("%d", &n);
    
    for(int i=0; i<n; i++)
        scanf("%d", &arr[i]);
        
    mergeSort(arr, 0, n-1);
    
    for(int i=0; i<n; i++)
        printf("%d ", arr[i]);
    return 0;
}

代碼的驗證

3. 應用

1. 求逆序對

題目鏈接

#include <iostream>

using namespace std;

const int N = 1e6+10;
int arr[N], tmp[N];

long long mergeSort(int *arr, int l, int r)
{
    if(l >= r) return 0;
    
    int mid = (l+r)>>1;
    long long ret = mergeSort(arr, l, mid) + mergeSort(arr, mid+1, r);
    
    int k = 0, i = l, j = mid+1;
    while(i<=mid && j<=r)
    {
        if(arr[i] <= arr[j])
            tmp[k++] = arr[i++];
        else{
            ret += mid-i+1;
            tmp[k++] = arr[j++];
        }
    }
    
    while(i <= mid) tmp[k++] = arr[i++];
    while(j <= r)   tmp[k++] = arr[j++];   
    
    for(int i=0, j=l; j<=r; i++,j++)    arr[j] = tmp[i];
    
    return ret;
}

int main()
{
    int n;
    scanf("%d", &n);
    
    for(int i=0; i<n; i++)  scanf("%d", &arr[i]);
    
    printf("%ld\n", mergeSort(arr, 0, n-1));
    
    //for(int i=0; i<n; i++) cout<<arr[i]<<" ";
    return 0;
}

• 統計時機：
  當arr[i] > arr[j]時，（mid-i+1）個數可與arr[j]構成逆序對。
  

• 注：本系列參考：AcWing。。。