# Check if array can be sorted by swapping pairs with GCD of set bits count equal to that of the smallest array element

• Last Updated : 01 Oct, 2021

Given an array arr[] consisting of N integers, the task is to check if it is possible to sort the array using the following swap operations:

Swapping of two numbers is valid only if the Greatest Common Divisor of count of set bits of the two numbers is equal to the number of set bits in the smallest element of the array.

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If it is possible to sort the array by performing only the above swaps, then print “Yes”. Otherwise, print “No”.

Examples:

Input: arr[] = {2, 3, 5, 7, 6}
Output: Yes
Explanation:
7 and 6 are needed to be swapped to make the array sorted.
7 has 3 set bits and 6 has 2 set bits.
Since GCD(2, 3) = 1, which is equal to the number of set bits in the smallest integer from the array i.e., 2 (1 set bit).
Therefore, the array can be sorted by swapping (7, 6).

Input: arr[] = {3, 3, 15, 7}
Output: No
Explanation:
15 and 7 are needed to be swapped to make the array sorted.
15 has 4 set bits and 7 has 3 set bits. GCD(4, 3) = 1, which is not equal to the number of set bits in the smallest integer from the array i.e., 3(2 set bit).
Therefore, the array cannot be sorted.

Approach: Follow the steps below to solve the problem:

1. Sort the given array and store it in an auxiliary array(say dup[]).
2. Iterate over the array and for every element, check if it is at the same index in both arr[] and dup[] or not. If found to be true, proceed to the next index.
3. Otherwise, if swapping of ith and jth position elements is required in arr[] then calculate the GCD of set bits of arr[i] with set bits of arr[j] and check if it is equal to the count of set bits in the smallest element of the array or not.
4. If any such swapping is not allowed, print “No”. Otherwise, print “Yes”.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// Function to count number of set``// bits in an integer``int` `calculateSetBit(``int` `n)``{``    ``int` `count = 0;` `    ``// Traverse every bits``    ``for` `(``int` `i = 0; i < 32; i++) {``        ``if` `(n & 1)``            ``count++;` `        ``// Right shift by 1``        ``n = n >> 1;``    ``}` `    ``return` `count;``}` `// Function to check if sorting the``// given array is possible or not``void` `sortPossible(``int` `arr[], ``int` `n)``{``    ``// Duplicate array``    ``int` `dup[n];` `    ``for` `(``int` `i = 0; i < n; i++)``        ``dup[i] = arr[i];` `    ``// Sorted array to check if the``    ``// original array can be sorted``    ``sort(dup, dup + n);` `    ``// Flag variable to check``    ``// if possible to sort``    ``bool` `flag = 1;` `    ``// Calculate bits of smallest``    ``// array element``    ``int` `bit = calculateSetBit(dup);` `    ``// Check every wrong placed``    ``// integer in the array``    ``for` `(``int` `i = 0; i < n; i++) {``        ``if` `(arr[i] != dup[i]) {` `            ``// Swapping only if GCD of set``            ``// bits is equal to set bits in``            ``// smallest integer``            ``if` `(__gcd(``                    ``calculateSetBit(arr[i]),``                    ``bit)``                ``!= bit) {``                ``flag = 0;``                ``break``;``            ``}``        ``}``    ``}` `    ``// Print the result``    ``if` `(flag) {``        ``cout << ``"Yes"``;``    ``}``    ``else` `{``        ``cout << ``"No"``;``    ``}` `    ``return``;``}` `// Driver Code``int` `main()``{``    ``int` `arr[] = { 3, 9, 12, 6 };` `    ``int` `N = ``sizeof``(arr) / ``sizeof``(arr);` `    ``// Function Call``    ``sortPossible(arr, N);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.io.*;``import` `java.util.*;` `class` `GFG{`` ` `// Recursive function to return``// gcd of a and b``static` `int` `gcd(``int` `a, ``int` `b)``{``    ` `    ``// Everything divides 0 ``    ``if` `(a == ``0``)``        ``return` `b;``    ``if` `(b == ``0``)``         ``return` `a;``       ` `    ``// Base case``    ``if` `(a == b)``        ``return` `a;``       ` `    ``// a is greater``    ``if` `(a > b)``        ``return` `gcd(a - b, b);``    ``return` `gcd(a, b - a);``}``    ` `// Function to count number of set``// bits in an integer``static` `int` `calculateSetBit(``int` `n)``{``    ``int` `count = ``0``;`` ` `    ``// Traverse every bits``    ``for``(``int` `i = ``0``; i < ``32``; i++)``    ``{``        ``if` `((n & ``1``) != ``0``)``            ``count++;`` ` `        ``// Right shift by 1``        ``n = n >> ``1``;``    ``}``    ``return` `count;``}`` ` `// Function to check if sorting the``// given array is possible or not``static` `void` `sortPossible(``int` `arr[], ``int` `n)``{``    ` `    ``// Duplicate array``    ``int` `dup[] = ``new` `int``[n];`` ` `    ``for``(``int` `i = ``0``; i < n; i++)``        ``dup[i] = arr[i];`` ` `    ``// Sorted array to check if the``    ``// original array can be sorted``    ``Arrays.sort(dup);`` ` `    ``// Flag variable to check``    ``// if possible to sort``    ``int` `flag = ``1``;`` ` `    ``// Calculate bits of smallest``    ``// array element``    ``int` `bit = calculateSetBit(dup[``0``]);`` ` `    ``// Check every wrong placed``    ``// integer in the array``    ``for``(``int` `i = ``0``; i < n; i++)``    ``{``        ``if` `(arr[i] != dup[i])``        ``{``            ` `            ``// Swapping only if GCD of set``            ``// bits is equal to set bits in``            ``// smallest integer``            ``if` `(gcd(calculateSetBit(``                ``arr[i]), bit) != bit)``            ``{``                ``flag = ``0``;``                ``break``;``            ``}``        ``}``    ``}`` ` `    ``// Print the result``    ``if` `(flag != ``0``)``    ``{``        ``System.out.println(``"Yes"``);``    ``}``    ``else``    ``{``        ``System.out.println(``"No"``);``    ``}``    ``return``;``}`` ` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `arr[] = { ``3``, ``9``, ``12``, ``6` `};`` ` `    ``int` `N = arr.length;`` ` `    ``// Function call``    ``sortPossible(arr, N);``}``}` `// This code is contributed by sanjoy_62`

## Python3

 `# Python3 program for the above approach``from` `math ``import` `gcd` `# Function to count number of set``# bits in an eger``def` `calculateSetBit(n):` `    ``count ``=` `0` `    ``# Traverse every bits``    ``for` `i ``in` `range``(``32``):``        ``if` `(n & ``1``):``            ``count ``+``=` `1` `        ``# Right shift by 1``        ``n ``=` `n >> ``1``        ` `    ``return` `count` `# Function to check if sorting the``# given array is possible or not``def` `sortPossible(arr, n):` `    ``# Duplicate array``    ``dup ``=` `[``0``] ``*` `n` `    ``for` `i ``in` `range``(n):``        ``dup[i] ``=` `arr[i]` `    ``# Sorted array to check if the``    ``# original array can be sorted``    ``dup ``=` `sorted``(dup)` `    ``# Flag variable to check``    ``# if possible to sort``    ``flag ``=` `1` `    ``# Calculate bits of smallest``    ``# array element``    ``bit ``=` `calculateSetBit(dup[``0``])` `    ``# Check every wrong placed``    ``# eger in the array``    ``for` `i ``in` `range``(n):``        ``if` `(arr[i] !``=` `dup[i]):` `            ``# Swapping only if GCD of set``            ``# bits is equal to set bits in``            ``# smallest eger``            ``if` `(gcd(calculateSetBit(arr[i]), bit) !``=` `bit):``                ``flag ``=` `0``                ``break` `    ``# Print the result``    ``if` `(flag):``        ``print``(``"Yes"``)``    ``else``:``        ``print``(``"No"``)` `    ``return` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``arr ``=` `[ ``3``, ``9``, ``12``, ``6` `]` `    ``N ``=` `len``(arr)` `    ``# Function call``    ``sortPossible(arr, N)` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program for the above approach ``using` `System;` `class` `GFG{`` ` `// Recursive function to return``// gcd of a and b``static` `int` `gcd(``int` `a, ``int` `b)``{``    ` `    ``// Everything divides 0 ``    ``if` `(a == 0)``        ``return` `b;``    ``if` `(b == 0)``         ``return` `a;``       ` `    ``// Base case``    ``if` `(a == b)``        ``return` `a;``       ` `    ``// a is greater``    ``if` `(a > b)``        ``return` `gcd(a - b, b);``    ``return` `gcd(a, b - a);``}``    ` `// Function to count number of set``// bits in an integer``static` `int` `calculateSetBit(``int` `n)``{``    ``int` `count = 0;`` ` `    ``// Traverse every bits``    ``for``(``int` `i = 0; i < 32; i++)``    ``{``        ``if` `((n & 1) != 0)``            ``count++;`` ` `        ``// Right shift by 1``        ``n = n >> 1;``    ``}``    ``return` `count;``}`` ` `// Function to check if sorting the``// given array is possible or not``static` `void` `sortPossible(``int``[] arr, ``int` `n)``{``    ` `    ``// Duplicate array``    ``int``[] dup = ``new` `int``[n];`` ` `    ``for``(``int` `i = 0; i < n; i++)``        ``dup[i] = arr[i];`` ` `    ``// Sorted array to check if the``    ``// original array can be sorted``    ``Array.Sort(dup);`` ` `    ``// Flag variable to check``    ``// if possible to sort``    ``int` `flag = 1;`` ` `    ``// Calculate bits of smallest``    ``// array element``    ``int` `bit = calculateSetBit(dup);`` ` `    ``// Check every wrong placed``    ``// integer in the array``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``if` `(arr[i] != dup[i])``        ``{``            ` `            ``// Swapping only if GCD of set``            ``// bits is equal to set bits in``            ``// smallest integer``            ``if` `(gcd(calculateSetBit(``                ``arr[i]), bit) != bit)``            ``{``                ``flag = 0;``                ``break``;``            ``}``        ``}``    ``}`` ` `    ``// Print the result``    ``if` `(flag != 0)``    ``{``        ``Console.WriteLine(``"Yes"``);``    ``}``    ``else``    ``{``        ``Console.WriteLine(``"No"``);``    ``}`` ` `    ``return``;``}`` ` `// Driver Code``public` `static` `void` `Main()``{``    ``int``[] arr = { 3, 9, 12, 6 };`` ` `    ``int` `N = arr.Length;`` ` `    ``// Function call``    ``sortPossible(arr, N);``}``}` `// This code is contributed by sanjoy_62`

## Javascript

 ``
Output:
`Yes`

Time Complexity: O(N)
Auxiliary Space: O(N)

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