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# Count of sub-arrays whose elements can be re-arranged to form palindromes

• Difficulty Level : Expert
• Last Updated : 27 Apr, 2021

Given an array arr[] of size n. The task is to count the number of possible sub-arrays such that their elements can be re-arranged to form a palindrome.
Examples:

Input: arr[] = {1, 2, 1, 2}
Output:
{1}, {2}, {1}, {2}, {1, 2, 1}, {2, 1, 2} and {1, 2, 1, 2} are the valid sub-arrays.
Input: arr[] = {1, 2, 3, 1, 2, 3, 4}
Output: 11

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Approach: There are a few observations:

• To create an even length palindrome all the distinct numbers need to have even occurrences.
• To create an odd length palindrome there has to be only one number of odd occurrences.

Now, the tricky part is to determine whether a particular section of the array can be made into a palindrome in O(1) complexity. We can use XOR to achieve this:

• For each number m, we can use it in the xor calculation as 2^n so that it contains a single set bit.
• If the xor of all the elements of a section is 0 then it means that occurrences of all the distinct numbers of this section are even.
• If the xor of all the elements of a section is greater than 0 then it means that:
• Either there is more than one distinct number with odd occurrences in which case the section cannot be re-arranged to form a palindrome
• Or exactly one number with odd occurrence (the binary representation of the number will have only 1 set bit).

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;``typedef` `signed` `long` `long` `ll;` `// Function that returns true if n is a``// power of 2 i.e. n has only 1 set bit``bool` `is_power_of_two(ll n)``{``    ``return` `!(n & (n - 1LL));``}` `// Function to return the count``// of all valid sub-arrays``int` `countSubArrays(``int` `arr[], ``int` `n)``{` `    ``// To store the count of valid sub-arrays``    ``int` `cnt = 0;` `    ``for` `(``int` `j = 0; j < n; j++) {``        ``ll xorval = 0LL;``        ``for` `(``int` `k = j; k < n; k++) {` `            ``// num = 2 ^ arr[k]``            ``ll num = 1LL << arr[k];``            ``xorval ^= num;` `            ``// If frequency of all the elements of the``            ``// sub-array is even or there is only a``            ``// single element with odd frequency``            ``if` `(xorval == 0LL || is_power_of_two(xorval))``                ``cnt++;``        ``}``    ``}` `    ``// Return the required count``    ``return` `cnt;``}` `// Driver code``int` `main()``{` `    ``int` `arr[] = { 1, 2, 3, 1, 2, 3, 4 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);``    ``cout << countSubArrays(arr, n);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``class` `GfG``{``    ` `static` `long` `ll;` `// Function that returns true if n is a``// power of 2 i.e. n has only 1 set bit``static` `boolean` `is_power_of_two(``long` `n)``{``    ``//return !(n & (n - 1));``    ``return` `false``;``}` `// Function to return the count``// of all valid sub-arrays``static` `int` `countSubArrays(``int` `arr[], ``int` `n)``{` `    ``// To store the count of valid sub-arrays``    ``int` `cnt = ``0``;` `    ``for` `(``int` `j = ``0``; j < n; j++)``    ``{``        ``long` `xorval = ``0``;``        ``for` `(``int` `k = j; k < n; k++)``        ``{` `            ``// num = 2 ^ arr[k]``            ``long` `num = ``1` `<< arr[k];``            ``xorval ^= num;` `            ``// If frequency of all the elements of the``            ``// sub-array is even or there is only a``            ``// single element with odd frequency``            ``if` `(xorval == ``0` `|| is_power_of_two(xorval))``                ``cnt++;``        ``}``    ``}` `    ``// Return the required count``    ``return` `cnt;``}` `// Driver code``public` `static` `void` `main(String[] args)``{` `    ``int` `arr[] = { ``1``, ``2``, ``3``, ``1``, ``2``, ``3``, ``4` `};``    ``int` `n = arr.length;``    ``System.out.println(countSubArrays(arr, n) + ``"1"``);``}``}` `// This code is contributed by Prerna Saini`

## Python3

 `# Python3 implementation of the approach` `# Function that returns true if n is a``# power of 2 i.e. n has only 1 set bit``def` `is_power_of_two(n):` `    ``return` `0` `if``(n & (n ``-` `1``)) ``else` `1``;` `# Function to return the count``# of all valid sub-arrays``def` `countSubArrays(arr, n):` `    ``# To store the count of valid sub-arrays``    ``cnt ``=` `0``;` `    ``for` `j ``in` `range``(n):``        ``xorval ``=` `0``;``        ``for` `k ``in` `range``(j, n):` `            ``# num = 2 ^ arr[k]``            ``num ``=` `1` `<< arr[k];``            ``xorval ^``=` `num;` `            ``# If frequency of all the elements of the``            ``# sub-array is even or there is only a``            ``# single element with odd frequency``            ``if` `(xorval ``=``=` `0` `or` `is_power_of_two(xorval)):``                ``cnt ``+``=` `1``;` `    ``# Return the required count``    ``return` `cnt;` `# Driver code``arr ``=` `[ ``1``, ``2``, ``3``, ``1``, ``2``, ``3``, ``4` `];``n ``=` `len``(arr);``print``(countSubArrays(arr, n));` `# This code is contributed by mits`

## C#

 `// C# implementation of the approach``using` `System;``    ` `class` `GfG``{``    ` `static` `long` `ll;` `// Function that returns true if n is a``// power of 2 i.e. n has only 1 set bit``static` `bool` `is_power_of_two(``long` `n)``{``    ``//return !(n & (n - 1));``    ``return` `false``;``}` `// Function to return the count``// of all valid sub-arrays``static` `int` `countSubArrays(``int` `[]arr, ``int` `n)``{` `    ``// To store the count of valid sub-arrays``    ``int` `cnt = 0;` `    ``for` `(``int` `j = 0; j < n; j++)``    ``{``        ``long` `xorval = 0;``        ``for` `(``int` `k = j; k < n; k++)``        ``{` `            ``// num = 2 ^ arr[k]``            ``long` `num = 1 << arr[k];``            ``xorval ^= num;` `            ``// If frequency of all the elements of the``            ``// sub-array is even or there is only a``            ``// single element with odd frequency``            ``if` `(xorval == 0 || is_power_of_two(xorval))``                ``cnt++;``        ``}``    ``}` `    ``// Return the required count``    ``return` `cnt;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{` `    ``int` `[]arr = { 1, 2, 3, 1, 2, 3, 4 };``    ``int` `n = arr.Length;``    ``Console.WriteLine(countSubArrays(arr, n) + ``"1"``);``}``}` `// This code contributed by Rajput-Ji`

## PHP

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## Javascript

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Output:
`11`

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