计数数组中的反转|设置 1（使用归并排序）

数组的反转计数表示 – 数组离排序的距离(或接近)。如果数组已经排序，则倒置计数为0，但如果数组按相反顺序排序，则倒置计数为最大值。
正式地说，如果 a[i] > a[j] 且 i < j，则两个元素 a[i] 和 a[j] 形成反转
例子：

Input: arr[] = {8, 4, 2, 1}
Output: 6

Explanation: Given array has six inversions:
(8, 4), (4, 2), (8, 2), (8, 1), (4, 1), (2, 1).


Input: arr[] = {3, 1, 2}
Output: 2

Explanation: Given array has two inversions:
(3, 1), (3, 2) 

方法一(简单)

• 方法：遍历数组，对于每一个索引，在其右边找到数组中较小元素的个数。这可以使用嵌套循环来完成。总结数组中所有索引的计数并打印总和。
• 算法：
  1. 从头到尾遍历数组
  2. 对于每个元素，使用另一个循环查找小于当前数量的元素计数，直到该索引为止。
  3. 总结每个索引的反转计数。
  4. 打印反转计数。
• 执行：

C++

// C++ program to Count Inversions
// in an array
#include 
using namespace std;
 
int getInvCount(int arr[], int n)
{
    int inv_count = 0;
    for (int i = 0; i < n - 1; i++)
        for (int j = i + 1; j < n; j++)
            if (arr[i] > arr[j])
                inv_count++;
 
    return inv_count;
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 20, 6, 4, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << " Number of inversions are "
         << getInvCount(arr, n);
    return 0;
}
 
// This code is contributed
// by Akanksha Rai

C

// C program to Count
// Inversions in an array
#include 
#include 
int getInvCount(int arr[], int n)
{
    int inv_count = 0;
    for (int i = 0; i < n - 1; i++)
        for (int j = i + 1; j < n; j++)
            if (arr[i] > arr[j])
                inv_count++;
 
    return inv_count;
}
 
/* Driver program to test above functions */
int main()
{
    int arr[] = { 1, 20, 6, 4, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
    printf(" Number of inversions are %d \n",
           getInvCount(arr, n));
    return 0;
}

Java

// Java program to count
// inversions in an array
class Test {
    static int arr[] = new int[] { 1, 20, 6, 4, 5 };
 
    static int getInvCount(int n)
    {
        int inv_count = 0;
        for (int i = 0; i < n - 1; i++)
            for (int j = i + 1; j < n; j++)
                if (arr[i] > arr[j])
                    inv_count++;
 
        return inv_count;
    }
 
    // Driver method to test the above function
    public static void main(String[] args)
    {
        System.out.println("Number of inversions are "
                           + getInvCount(arr.length));
    }
}

Python3

# Python3 program to count
# inversions in an array
 
 
def getInvCount(arr, n):
 
    inv_count = 0
    for i in range(n):
        for j in range(i + 1, n):
            if (arr[i] > arr[j]):
                inv_count += 1
 
    return inv_count
 
 
# Driver Code
arr = [1, 20, 6, 4, 5]
n = len(arr)
print("Number of inversions are",
      getInvCount(arr, n))
 
# This code is contributed by Smitha Dinesh Semwal

C#

// C# program to count inversions
// in an array
using System;
using System.Collections.Generic;
 
class GFG {
 
    static int[] arr = new int[] { 1, 20, 6, 4, 5 };
 
    static int getInvCount(int n)
    {
        int inv_count = 0;
 
        for (int i = 0; i < n - 1; i++)
            for (int j = i + 1; j < n; j++)
                if (arr[i] > arr[j])
                    inv_count++;
 
        return inv_count;
    }
 
    // Driver code
    public static void Main()
    {
        Console.WriteLine("Number of "
                          + "inversions are "
                          + getInvCount(arr.Length));
    }
}
 
// This code is contributed by Sam007

PHP

 $arr[$j])
                $inv_count++;
 
    return $inv_count;
}
 
// Driver Code
$arr = array(1, 20, 6, 4, 5 );
$n = sizeof($arr);
echo "Number of inversions are ",
           getInvCount($arr, $n);
 
// This code is contributed by ita_c
?>

Javascript

// Javascript program to count inversions in an array
 
arr = [1, 20, 6, 4, 5];
 
function getInvCount(arr){
    let inv_count = 0;
    for(let i=0; i<arr.length-1; i++){
        for(let j=i+1; j<arr.length; j++){
            if(arr[i] > arr[j]) inv_count++;
        }
    }
    return inv_count;
}
 
// function call
document.write("Number of inversions are "+ getInvCount(arr));
 
// This code is contributed by Karthik SP

输出

Number of inversions are 5

• 复杂度分析：
  • 时间复杂度： O(n^2)，从头到尾遍历数组需要两个嵌套循环，所以时间复杂度为O(n^2)
  • 空间复杂度： O(1)，不需要额外的空间。

方法二(增强归并排序)

• 方法：
  假设数组左半部分和右半部分的反转次数(假设为 inv1 和 inv2)； Inv1 + Inv2 中没有考虑哪些类型的反转?答案是——在合并步骤中需要计算的反转。因此，要得到需要添加的反演总数是左子数组、右子数组和merge()中的反演次数。

• 如何获得merge()中的反转次数?
  在合并过程中，让 i 用于索引左子数组，j 用于右子数组。在merge() 的任何步骤中，如果a[i] 大于a[j]，则存在(mid – i) 次反转。因为左右子数组是有序的，所以左子数组(a[i+1], a[i+2] … a[mid])中剩余的所有元素都会大于a[j]

• 完整图：

• 算法：
  1. 这个想法类似于归并排序，在每一步中将数组分成相等或几乎相等的两半，直到达到基本情况。
  2. 创建一个函数merge 来计算数组的两半合并时的反转次数，创建两个索引 i 和 j，i 是前半部分的索引，j 是后半部分的索引。如果 a[i] 大于 a[j]，则存在 (mid – i) 个反转。因为左右子数组是有序的，所以左子数组(a[i+1], a[i+2] … a[mid])中的所有剩余元素都会大于a[j]。
  3. 创建一个递归函数将数组分成两半，并通过求和求前半部分的反转次数、后半部分的反转次数和合并两者的反转次数来找到答案。
  4. 递归的基本情况是给定的一半中只有一个元素。
  5. 打印答案
• 执行：

C++

// C++ program to Count
// Inversions in an array
// using Merge Sort
#include 
using namespace std;
 
int _mergeSort(int arr[], int temp[], int left, int right);
int merge(int arr[], int temp[], int left, int mid,
          int right);
 
/* This function sorts the
   input array and returns the
number of inversions in the array */
int mergeSort(int arr[], int array_size)
{
    int temp[array_size];
    return _mergeSort(arr, temp, 0, array_size - 1);
}
 
/* An auxiliary recursive function
  that sorts the input array and
returns the number of inversions in the array. */
int _mergeSort(int arr[], int temp[], int left, int right)
{
    int mid, inv_count = 0;
    if (right > left) {
        /* Divide the array into two parts and
        call _mergeSortAndCountInv()
        for each of the parts */
        mid = (right + left) / 2;
 
        /* Inversion count will be sum of
        inversions in left-part, right-part
        and number of inversions in merging */
        inv_count += _mergeSort(arr, temp, left, mid);
        inv_count += _mergeSort(arr, temp, mid + 1, right);
 
        /*Merge the two parts*/
        inv_count += merge(arr, temp, left, mid + 1, right);
    }
    return inv_count;
}
 
/* This funt merges two sorted arrays
and returns inversion count in the arrays.*/
int merge(int arr[], int temp[], int left, int mid,
          int right)
{
    int i, j, k;
    int inv_count = 0;
 
    i = left; /* i is index for left subarray*/
    j = mid; /* j is index for right subarray*/
    k = left; /* k is index for resultant merged subarray*/
    while ((i <= mid - 1) && (j <= right)) {
        if (arr[i] <= arr[j]) {
            temp[k++] = arr[i++];
        }
        else {
            temp[k++] = arr[j++];
 
            /* this is tricky -- see above
            explanation/diagram for merge()*/
            inv_count = inv_count + (mid - i);
        }
    }
 
    /* Copy the remaining elements of left subarray
(if there are any) to temp*/
    while (i <= mid - 1)
        temp[k++] = arr[i++];
 
    /* Copy the remaining elements of right subarray
       (if there are any) to temp*/
    while (j <= right)
        temp[k++] = arr[j++];
 
    /*Copy back the merged elements to original array*/
    for (i = left; i <= right; i++)
        arr[i] = temp[i];
 
    return inv_count;
}
 
// Driver code
int main()
{
    int arr[] = { 1, 20, 6, 4, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int ans = mergeSort(arr, n);
    cout << " Number of inversions are " << ans;
    return 0;
}
 
// This is code is contributed by rathbhupendra

C

// C program to Count
// Inversions in an array
// using Merge Sort
#include 
#include 
 
int _mergeSort(int arr[], int temp[],
               int left, int right);
int merge(int arr[], int temp[],
          int left, int mid,
          int right);
 
/* This function sorts the input array and returns the
   number of inversions in the array */
int mergeSort(int arr[], int array_size)
{
    int* temp = (int*)malloc(sizeof(int) * array_size);
    return _mergeSort(arr, temp, 0,
                      array_size - 1);
}
 
/* An auxiliary recursive function
   that sorts the input
  array and returns the number
  of inversions in the array.
*/
int _mergeSort(int arr[], int temp[], int left, int right)
{
    int mid, inv_count = 0;
    if (right > left)
    {
        /* Divide the array into two parts and call
       _mergeSortAndCountInv() for each of the parts */
        mid = (right + left) / 2;
 
        /* Inversion count will be the sum of inversions in
        left-part, right-part and number of inversions in
        merging */
        inv_count += _mergeSort(arr, temp, left, mid);
        inv_count += _mergeSort(arr, temp, mid + 1, right);
 
        /*Merge the two parts*/
        inv_count += merge(arr, temp, left, mid + 1, right);
    }
    return inv_count;
}
 
/* This funt merges two sorted
   arrays and returns inversion
   count in the arrays.*/
int merge(int arr[], int temp[], int left, int mid,
          int right)
{
    int i, j, k;
    int inv_count = 0;
 
    i = left; /* i is index for left subarray*/
    j = mid; /* j is index for right subarray*/
    k = left; /* k is index for resultant merged subarray*/
    while ((i <= mid - 1) && (j <= right)) {
        if (arr[i] <= arr[j]) {
            temp[k++] = arr[i++];
        }
        else
        {
            temp[k++] = arr[j++];
 
            /*this is tricky -- see above
             * explanation/diagram for merge()*/
            inv_count = inv_count + (mid - i);
        }
    }
 
    /* Copy the remaining elements of left subarray
   (if there are any) to temp*/
    while (i <= mid - 1)
        temp[k++] = arr[i++];
 
    /* Copy the remaining elements of right subarray
   (if there are any) to temp*/
    while (j <= right)
        temp[k++] = arr[j++];
 
    /*Copy back the merged elements to original array*/
    for (i = left; i <= right; i++)
        arr[i] = temp[i];
 
    return inv_count;
}
 
/* Driver code*/
int main(int argv, char** args)
{
    int arr[] = { 1, 20, 6, 4, 5 };
    printf(" Number of inversions are %d \n",
           mergeSort(arr, 5));
    getchar();
    return 0;
}

Java

// Java implementation of the approach
import java.util.Arrays;
 
public class GFG {
 
    // Function to count the number of inversions
    // during the merge process
    private static int mergeAndCount(int[] arr, int l,
                                     int m, int r)
    {
 
        // Left subarray
        int[] left = Arrays.copyOfRange(arr, l, m + 1);
 
        // Right subarray
        int[] right = Arrays.copyOfRange(arr, m + 1, r + 1);
 
        int i = 0, j = 0, k = l, swaps = 0;
 
        while (i < left.length && j < right.length) {
            if (left[i] <= right[j])
                arr[k++] = left[i++];
            else {
                arr[k++] = right[j++];
                swaps += (m + 1) - (l + i);
            }
        }
        while (i < left.length)
            arr[k++] = left[i++];
        while (j < right.length)
            arr[k++] = right[j++];
        return swaps;
    }
 
    // Merge sort function
    private static int mergeSortAndCount(int[] arr, int l,
                                         int r)
    {
 
        // Keeps track of the inversion count at a
        // particular node of the recursion tree
        int count = 0;
 
        if (l < r) {
            int m = (l + r) / 2;
 
            // Total inversion count = left subarray count
            // + right subarray count + merge count
 
            // Left subarray count
            count += mergeSortAndCount(arr, l, m);
 
            // Right subarray count
            count += mergeSortAndCount(arr, m + 1, r);
 
            // Merge count
            count += mergeAndCount(arr, l, m, r);
        }
 
        return count;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int[] arr = { 1, 20, 6, 4, 5 };
 
        System.out.println(
            mergeSortAndCount(arr, 0, arr.length - 1));
    }
}
 
// This code is contributed by Pradip Basak

蟒蛇3

# Python 3 program to count inversions in an array
 
# Function to Use Inversion Count
def mergeSort(arr, n):
    # A temp_arr is created to store
    # sorted array in merge function
    temp_arr = [0]*n
    return _mergeSort(arr, temp_arr, 0, n-1)
 
# This Function will use MergeSort to count inversions
 
def _mergeSort(arr, temp_arr, left, right):
 
    # A variable inv_count is used to store
    # inversion counts in each recursive call
 
    inv_count = 0
 
    # We will make a recursive call if and only if
    # we have more than one elements
 
    if left < right:
 
        # mid is calculated to divide the array into two subarrays
        # Floor division is must in case of python
 
        mid = (left + right)//2
 
        # It will calculate inversion
        # counts in the left subarray
 
        inv_count += _mergeSort(arr, temp_arr,
                                    left, mid)
 
        # It will calculate inversion
        # counts in right subarray
 
        inv_count += _mergeSort(arr, temp_arr,
                                  mid + 1, right)
 
        # It will merge two subarrays in
        # a sorted subarray
 
        inv_count += merge(arr, temp_arr, left, mid, right)
    return inv_count
 
# This function will merge two subarrays
# in a single sorted subarray
def merge(arr, temp_arr, left, mid, right):
    i = left     # Starting index of left subarray
    j = mid + 1 # Starting index of right subarray
    k = left     # Starting index of to be sorted subarray
    inv_count = 0
 
    # Conditions are checked to make sure that
    # i and j don't exceed their
    # subarray limits.
 
    while i <= mid and j <= right:
 
        # There will be no inversion if arr[i] <= arr[j]
 
        if arr[i] <= arr[j]:
            temp_arr[k] = arr[i]
            k += 1
            i += 1
        else:
            # Inversion will occur.
            temp_arr[k] = arr[j]
            inv_count += (mid-i + 1)
            k += 1
            j += 1
 
    # Copy the remaining elements of left
    # subarray into temporary array
    while i <= mid:
        temp_arr[k] = arr[i]
        k += 1
        i += 1
 
    # Copy the remaining elements of right
    # subarray into temporary array
    while j <= right:
        temp_arr[k] = arr[j]
        k += 1
        j += 1
 
    # Copy the sorted subarray into Original array
    for loop_var in range(left, right + 1):
        arr[loop_var] = temp_arr[loop_var]
         
    return inv_count
 
# Driver Code
# Given array is
arr = [1, 20, 6, 4, 5]
n = len(arr)
result = mergeSort(arr, n)
print("Number of inversions are", result)
 
# This code is contributed by ankush_953

C＃

// C# implementation of counting the
// inversion using merge sort
 
using System;
public class Test {
 
    /* This method sorts the input array and returns the
       number of inversions in the array */
    static int mergeSort(int[] arr, int array_size)
    {
        int[] temp = new int[array_size];
        return _mergeSort(arr, temp, 0, array_size - 1);
    }
 
    /* An auxiliary recursive method that sorts the input
      array and returns the number of inversions in the
      array. */
    static int _mergeSort(int[] arr, int[] temp, int left,
                          int right)
    {
        int mid, inv_count = 0;
        if (right > left) {
            /* Divide the array into two parts and call
           _mergeSortAndCountInv() for each of the parts */
            mid = (right + left) / 2;
 
            /* Inversion count will be the sum of inversions
          in left-part, right-part
          and number of inversions in merging */
            inv_count += _mergeSort(arr, temp, left, mid);
            inv_count
                += _mergeSort(arr, temp, mid + 1, right);
 
            /*Merge the two parts*/
            inv_count
                += merge(arr, temp, left, mid + 1, right);
        }
        return inv_count;
    }
 
    /* This method merges two sorted arrays and returns
       inversion count in the arrays.*/
    static int merge(int[] arr, int[] temp, int left,
                     int mid, int right)
    {
        int i, j, k;
        int inv_count = 0;
 
        i = left; /* i is index for left subarray*/
        j = mid; /* j is index for right subarray*/
        k = left; /* k is index for resultant merged
                     subarray*/
        while ((i <= mid - 1) && (j <= right)) {
            if (arr[i] <= arr[j]) {
                temp[k++] = arr[i++];
            }
            else {
                temp[k++] = arr[j++];
 
                /*this is tricky -- see above
                 * explanation/diagram for merge()*/
                inv_count = inv_count + (mid - i);
            }
        }
 
        /* Copy the remaining elements of left subarray
       (if there are any) to temp*/
        while (i <= mid - 1)
            temp[k++] = arr[i++];
 
        /* Copy the remaining elements of right subarray
       (if there are any) to temp*/
        while (j <= right)
            temp[k++] = arr[j++];
 
        /*Copy back the merged elements to original array*/
        for (i = left; i <= right; i++)
            arr[i] = temp[i];
 
        return inv_count;
    }
 
    // Driver method to test the above function
    public static void Main()
    {
        int[] arr = new int[] { 1, 20, 6, 4, 5 };
        Console.Write("Number of inversions are "
                      + mergeSort(arr, 5));
    }
}
// This code is contributed by Rajput-Ji

Javascript

    // Function to count the number of inversions
    // during the merge process
    function mergeAndCount(arr,l,m,r)
    {
     
        // Left subarray
        let left = [];
        for(let i = l; i < m + 1; i++)
        {
            left.push(arr[i]);
             
        }
         
        // Right subarray
        let right = [];
        for(let i = m + 1; i < r + 1; i++)
        {
            right.push(arr[i]);
        }
        let i = 0, j = 0, k = l, swaps = 0;
        while (i < left.length && j < right.length)
        {
            if (left[i] <= right[j])
            {
                arr[k++] = left[i++];
            }
            else
            {
                arr[k++] = right[j++];
                swaps += (m + 1) - (l + i);
            }
        }
        while (i < left.length)
        {
            arr[k++] = left[i++];
        }
        while (j < right.length)
        {
            arr[k++] = right[j++];
        }
        return swaps;
    }
     
    // Merge sort function
    function mergeSortAndCount(arr, l, r)
    {
         
        // Keeps track of the inversion count at a
        // particular node of the recursion tree
        let count = 0;
        if (l < r)
        {
            let m = Math.floor((l + r) / 2);
             
            // Total inversion count = left subarray count
            // + right subarray count + merge count
             
            // Left subarray count
            count += mergeSortAndCount(arr, l, m);
             
            // Right subarray count
            count += mergeSortAndCount(arr, m + 1, r);
             
            // Merge count
            count += mergeAndCount(arr, l, m, r);
        }
        return count;
    }
     
    // Driver code
    let arr= new Array(1, 20, 6, 4, 5 );
    document.write(mergeSortAndCount(arr, 0, arr.length - 1));
     
    // This code is contributed by avanitrachhadiya2155

输出：

Number of inversions are 5

复杂度分析：

• 时间复杂度： O(n log n)，采用的算法是分治法，所以每一层都需要一次全数组遍历，有log n层，所以时间复杂度是O(n log n)。
• 空间复杂度： O(n)，临时数组。

请注意，上面的代码修改(或排序)输入数组。如果我们只想计算反转，我们需要创建原始数组的副本并在副本上调用 mergeSort() 以保留原始数组的顺序。

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