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Sorting possible using size 3 subarray rotation

  • Difficulty Level : Medium
  • Last Updated : 23 Jun, 2021

Given an array of integer values which need to be sorted by only one operation – subarray rotation where subarray size should be 3. For example, If our array is (1 2 3 4) then we can reach to (1 4 2 3), (3 1 2 4) in one step. We need to tell whether it is possible to sort the complete array with this operation or not.
Examples : 
 

Input  : arr[] = [1, 3, 4, 2]
Output : Yes
Possible by below rotations, 
[1, 3, 4, 2] -> [1, 4, 2, 3] -> 
[1, 2, 3, 4]

Input  : arr[] = [1, 2, 4, 3]
Output : No
Not possible to sort above array by any 3 
size subarray rotation.

 

Suppose we have one subarray as [A[i] A[i+1] A[i+2]]. After one rotation, we get [A[i+2], A[i], A[i+1]] 
If we observe inversions before and after rotation, we can see that parity of inversion is not changed i.e. if [A[i] A[i+1] A[i+2]] has even number of inversions [A[i+2] A[i] A[i+1]] will have even inversions. Same is true for odd inversions. Due to movement of A[i+2], inversions either increase by 2 or decrease by 2 or remain same i.e. their parity won’t change. 
After observing above fact we can say that if initial array configuration has even number of inversion then it is possible to make them zero making array completely sorted otherwise not. We use merge sort based method for counting inversions. After getting the number of inversions, we can easily check parity of inversion and conclude whether it is possible to sort the array or not. 
 

C++




// C++ program to check whether we can sort
// given array using 3 size subarray rotation
// or not
#include <bits/stdc++.h>
using namespace std;
 
/* This function 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;
}
 
/* 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 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);
}
 
// method returns true is array can be sorted by 3
// size subarray rotation
bool possibleSortingBy3SizeSubarray(int arr[], int N)
{
    int numberOfInversion = mergeSort(arr, N);
 
    // if number of inversions are even then only
    // we can sort the array
    return (numberOfInversion % 2 == 0);
}
 
//  Driver code to test above methods
int main()
{
    int arr[] = {1, 3, 4, 2};
    int N = sizeof(arr) / sizeof(int);
 
    possibleSortingBy3SizeSubarray(arr, N)?
        cout << "Yes\n" : cout << "No\n";
}

Java




// Java program to check whether we can sort
// given array using 3 size subarray rotation
// or not
import java.io.*;
 
class GFG
{
    /* This function merges two sorted arrays and
    returns inversion count in the arrays.*/
    public static int merge(int[] arr, int left, int mid, int right)
    {
        int[] temp = new int[arr.length];
        int inv_count = 0;
        int i = left; /* i is index for left subarray*/
        int j = mid; /* j is index for right subarray*/
        int k = left; /* k is index for resultant merged
                         subarray*/
         
        while((i <= mid-1) && (j <= right))
        {
            if(arr[i] <= arr[j])
            {
                temp[k++] = arr[i];
                i++;
            }
            else
            {
                temp[k++] = arr[j];
                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 (int l = left; l <= right; l++)
            arr[l] = temp[l];
  
        return inv_count;
    }
     
    /* An auxiliary recursive function that sorts
    the input array and returns the number of
    inversions in the array. */
    public static int _mergeSort(int[] arr, int left, int right)
    {
        
       int mid, inv_count = 0;
       if(left < right)
       {
            
            /* Divide the array into two parts and
            call _mergeSortAndCountInv() for each
            of the parts */
            mid = (left + right)/2;
            
            /* Inversion count will be sum of inversions
            in left-part, right-part and number of
            inversions in merging */
            inv_count = _mergeSort(arr, left, mid);
            inv_count += _mergeSort(arr, mid+1, right);
            
            inv_count += merge(arr, left, mid+1, right);
            
        }
       return inv_count;
        
   }
     
    /* This function sorts the input array and returns the
    number of inversions in the array */
    public static int mergeSort(int[] arr, int N)
    {
     
        return _mergeSort(arr, 0, N-1);
    }
     
    public static boolean possibleSortingBy3SizeSubarray(int arr[], int N)
    {
        int numberOfInversion = mergeSort(arr, N);
  
        // if number of inversions are even then only
        // we can sort the array
        return (numberOfInversion % 2 == 0);
    }
     
    //  Driver code to test above methods
    public static void main (String[] args)
    {
         
        int arr[] = {1, 3, 4, 2};
        int N = arr.length;
  
        if(possibleSortingBy3SizeSubarray(arr, N))
            System.out.println( "Yes");
        else
            System.out.println("No");
    }
}

Python3




# Python3 program to check whether we can sort
# given array using 3 size subarray rotation or not
 
# This function merges two sorted arrays and
# returns inversion count in the arrays.
def merge(arr, temp, left, mid, right):
     
    # i is index for left subarray
    # j is index for right subarray
    # k is index for resultant merged subarray
    i, j, k, inv_count = left, mid, left, 0
     
    while (i <= mid - 1) and (j <= right):
     
        if arr[i] <= arr[j]:
            temp[k] = arr[i]
            k, i = k + 1, i + 1
         
        else:
            temp[k] = arr[j]
            k, j = k + 1, j + 1
 
            # 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]
        k, i = k + 1, i + 1
 
    # Copy the remaining elements of right
    # subarray (if there are any) to temp
    while j <= right:
        temp[k] = arr[j]
        k, j = k + 1, j + 1
 
    # Copy back the merged elements
    # to original array
    for i in range(left, right + 1):
        arr[i] = temp[i]
 
    return inv_count
 
# An auxiliary recursive function that
# sorts the input array and returns the
# number of inversions in the array.
def _mergeSort(arr, temp, left, right):
 
    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 function sorts the input array and
# returns the number of inversions in the array
def mergeSort(arr, array_size):
 
    temp = [None] * array_size
    return _mergeSort(arr, temp, 0, array_size - 1)
 
# method returns true is array can be
# sorted by 3 size subarray rotation
def possibleSortingBy3SizeSubarray(arr, N):
 
    numberOfInversion = mergeSort(arr, N)
 
    # if number of inversions are even
    # then only we can sort the array
    return (numberOfInversion % 2 == 0)
 
# Driver Code
if __name__ == "__main__":
 
    arr = [1, 3, 4, 2]
    N = len(arr)
 
    if possibleSortingBy3SizeSubarray(arr, N):
        print("Yes")
    else:
        print("No")
 
# This code is contributed by Rituraj Jain

C#




// C# program to check whether we
// can sort given array using 3 size
// subarray rotation or not.
using System;
 
class GFG
{
    /* This function merges two sorted arrays and
       returns inversion count in the arrays.*/
    public static int merge(int []arr, int left,
                             int mid, int right)
    {
        int []temp = new int[arr.Length];
        int inv_count = 0;
         
        /* i is index for left subarray*/
        int i = left;
         
        /* j is index for right subarray*/
        int j = mid; 
         
        /* k is index for resultant merged subarray*/
        int k = left;
         
        while((i <= mid-1) && (j <= right))
        {
            if(arr[i] <= arr[j])
            {
                temp[k++] = arr[i];
                i++;
            }
            else
            {
                temp[k++] = arr[j];
                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 (int l = left; l <= right; l++)
            arr[l] = temp[l];
 
        return inv_count;
    }
     
    /* An auxiliary recursive function that sorts
       the input array and returns the number of
       inversions in the array. */
    public static int _mergeSort(int []arr, int left,
                                           int right)
    {
      int mid, inv_count = 0;
      if(left < right)
      {
            /* Divide the array into two parts and
               call _mergeSortAndCountInv() for each
               of the parts */
            mid = (left + right)/2;
             
            /* Inversion count will be sum of inversions
               in left-part, right-part and number of
               inversions in merging */
            inv_count = _mergeSort(arr, left, mid);
            inv_count += _mergeSort(arr, mid+1, right);
            inv_count += merge(arr, left, mid+1, right);
             
     }
    return inv_count;
         
}
     
    /* This function sorts the input array
       and returns the number of inversions
       in the array */
    public static int mergeSort(int[] arr, int N)
    {
        return _mergeSort(arr, 0, N-1);
    }
     
    public static bool possibleSortingBy3SizeSubarray(int []arr,
                                                          int N)
    {
        int numberOfInversion = mergeSort(arr, N);
 
        // if number of inversions are even
        // then only we can sort the array
        return (numberOfInversion % 2 == 0);
    }
     
    // Driver code to test above methods
    public static void Main ()
    {
        int []arr = {1, 3, 4, 2};
        int N = arr.Length;
 
        if(possibleSortingBy3SizeSubarray(arr, N))
            Console.Write( "Yes");
        else
            Console.Write("No");
    }
}
 
// This code is contributed by nitin mittal.

Javascript




<script>
 
// JavaScript program to check whether we can sort
// given array using 3 size subarray rotation
// or not
 
 
/* This function merges two sorted arrays and
returns inversion count in the arrays.*/
function merge(arr, left, mid, right) {
    let temp = new Array(arr.length);
    let inv_count = 0;
    let i = left; /* i is index for left subarray*/
    let j = mid; /* j is index for right subarray*/
    let k = left; /* k is index for resultant merged
                        subarray*/
 
    while ((i <= mid - 1) && (j <= right)) {
        if (arr[i] <= arr[j]) {
            temp[k++] = arr[i];
            i++;
        }
        else {
            temp[k++] = arr[j];
            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 (let l = left; l <= right; l++)
        arr[l] = temp[l];
 
    return inv_count;
}
 
/* An auxiliary recursive function that sorts
the input array and returns the number of
inversions in the array. */
function _mergeSort(arr, left, right) {
 
    let mid, inv_count = 0;
    if (left < right) {
 
        /* Divide the array into two parts and
        call _mergeSortAndCountInv() for each
        of the parts */
        mid = Math.floor((left + right) / 2);
 
        /* Inversion count will be sum of inversions
        in left-part, right-part and number of
        inversions in merging */
        inv_count = _mergeSort(arr, left, mid);
        inv_count += _mergeSort(arr, mid + 1, right);
 
        inv_count += merge(arr, left, mid + 1, right);
 
    }
    return inv_count;
 
}
 
/* This function sorts the input array and returns the
number of inversions in the array */
function mergeSort(arr, N) {
 
    return _mergeSort(arr, 0, N - 1);
}
 
function possibleSortingBy3SizeSubarray(arr, N) {
    let numberOfInversion = mergeSort(arr, N);
 
    // if number of inversions are even then only
    // we can sort the array
    return (numberOfInversion % 2 == 0);
}
 
// Driver code to test above methods
 
 
let arr = [1, 3, 4, 2];
let N = arr.length;
 
if (possibleSortingBy3SizeSubarray(arr, N))
    document.write("Yes");
else
    document.write("No");
 
</script>

Output:  

Yes

Time Complexity : O(n Log n)
This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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