# Sum of bitwise AND of all subarrays

Given an array consisting of N positive integers, find the sum of bit-wise and of all possible sub-arrays of the array.

Examples:

```Input : arr[] = {1, 5, 8}
Output : 15
Bit-wise AND of {1} = 1
Bit-wise AND of {1, 5} = 1
Bit-wise AND of {1, 5, 8} = 0
Bit-wise AND of {5} = 5
Bit-wise AND of {5, 8} = 0
Bit-wise AND of {8} = 8

Sum = 1 + 1 + 0 + 5 + 0 + 8 =  15

Input : arr[] =  {7, 1, 1, 5}
Output : 20
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Simple Solution: A simple solution will be to generate all the sub-arrays, and sum up the AND values of all the sub-arrays. It will take linear time on an average to find the AND value of a sub-array and thus, the over all time complexity will be O(n3).

Efficient Solution: For the sake of better understanding, let’s assume that any bit of an element is represented by the variable ‘i’ and the variable ‘sum’ is used to store the final sum.

The idea here is, we will try to find the number of AND values(sub-arrays with bit-wise and(&)) with ith bit set. Let us suppose, there are ‘Si‘ number of sub-arrays with ith bit set. For, ith bit, sum can be updated as sum += (2i * S).

We will break the task to multiple steps. At each step, we will try to find the number of AND values with ith bit set. For this, we will simply iterate through the array and find the number of contiguous segments with ith bit set and there lengths. For, each such segment of length ‘l’, value of sum can be updated as sum += (2i * l * (l + 1))/2.

Since, for each bit, we are performing O(N) iterations and as there are at most log(max(A)) bits, the time complexity of this approach will be O(N*log(max(A)), assuming max(A) = maximum value in the array.

Below is the implementation of the above idea:

## C++

 `// CPP program to find sum of bitwise AND ` `// of all subarrays ` ` `  `#include ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find the sum of ` `// bitwise AND of all subarrays ` `int` `findAndSum(``int` `arr[], ``int` `n) ` `{ ` `    ``// variable to store ` `    ``// the final sum ` `    ``int` `sum = 0; ` ` `  `    ``// multiplier ` `    ``int` `mul = 1; ` ` `  `    ``for` `(``int` `i = 0; i < 30; i++) { ` `        ``// variable to check if ` `        ``// counting is on ` `        ``bool` `count_on = 0; ` ` `  `        ``// variable to store the ` `        ``// length of the subarrays ` `        ``int` `l = 0; ` ` `  `        ``// loop to find the contiguous ` `        ``// segments ` `        ``for` `(``int` `j = 0; j < n; j++) { ` `            ``if` `((arr[j] & (1 << i)) > 0) ` `                ``if` `(count_on) ` `                    ``l++; ` `                ``else` `{ ` `                    ``count_on = 1; ` `                    ``l++; ` `                ``} ` ` `  `            ``else` `if` `(count_on) { ` `                ``sum += ((mul * l * (l + 1)) / 2); ` `                ``count_on = 0; ` `                ``l = 0; ` `            ``} ` `        ``} ` ` `  `        ``if` `(count_on) { ` `            ``sum += ((mul * l * (l + 1)) / 2); ` `            ``count_on = 0; ` `            ``l = 0; ` `        ``} ` ` `  `        ``// updating the multiplier ` `        ``mul *= 2; ` `    ``} ` ` `  `    ``// returning the sum ` `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` ` `  `    ``int` `arr[] = { 7, 1, 1, 5 }; ` ` `  `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` ` `  `    ``cout << findAndSum(arr, n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to find sum of bitwise AND ` `// of all subarrays ` `class` `GFG ` `{ ` `     `  `// Function to find the sum of ` `// bitwise AND of all subarrays ` `static` `int` `findAndSum(``int` `[]arr, ``int` `n) ` `{ ` `    ``// variable to store ` `    ``// the final sum ` `    ``int` `sum = ``0``; ` ` `  `    ``// multiplier ` `    ``int` `mul = ``1``; ` ` `  `    ``for` `(``int` `i = ``0``; i < ``30``; i++)  ` `    ``{ ` `        ``// variable to check if ` `        ``// counting is on ` `        ``boolean` `count_on = ``false``; ` ` `  `        ``// variable to store the ` `        ``// length of the subarrays ` `        ``int` `l = ``0``; ` ` `  `        ``// loop to find the contiguous ` `        ``// segments ` `        ``for` `(``int` `j = ``0``; j < n; j++)  ` `        ``{ ` `            ``if` `((arr[j] & (``1` `<< i)) > ``0``) ` `                ``if` `(count_on) ` `                    ``l++; ` `                ``else` `                ``{ ` `                    ``count_on = ``true``; ` `                    ``l++; ` `                ``} ` ` `  `            ``else` `if` `(count_on) ` `            ``{ ` `                ``sum += ((mul * l * (l + ``1``)) / ``2``); ` `                ``count_on = ``false``; ` `                ``l = ``0``; ` `            ``} ` `        ``} ` ` `  `        ``if` `(count_on)  ` `        ``{ ` `            ``sum += ((mul * l * (l + ``1``)) / ``2``); ` `            ``count_on = ``false``; ` `            ``l = ``0``; ` `        ``} ` ` `  `        ``// updating the multiplier ` `        ``mul *= ``2``; ` `    ``} ` ` `  `    ``// returning the sum ` `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `[]arr = { ``7``, ``1``, ``1``, ``5` `}; ` `    ``int` `n = arr.length; ` ` `  `    ``System.out.println(findAndSum(arr, n)); ` `} ` `} ` ` `  `// This code is contributed  ` `// by Code_Mech. `

## Python3

 `# Python3 program to find Sum of  ` `# bitwise AND of all subarrays ` `import` `math as mt ` ` `  `# Function to find the Sum of ` `# bitwise AND of all subarrays ` `def` `findAndSum(arr, n): ` `     `  `    ``# variable to store the final Sum ` `    ``Sum` `=` `0` ` `  `    ``# multiplier ` `    ``mul ``=` `1` ` `  `    ``for` `i ``in` `range``(``30``): ` `         `  `        ``# variable to check if counting is on ` `        ``count_on ``=` `0` ` `  `        ``# variable to store the length ` `        ``# of the subarrays ` `        ``l ``=` `0` ` `  `        ``# loop to find the contiguous ` `        ``# segments ` `        ``for` `j ``in` `range``(n): ` ` `  `            ``if` `((arr[j] & (``1` `<< i)) > ``0``): ` `                ``if` `(count_on): ` `                    ``l ``+``=` `1` `                ``else``: ` `                    ``count_on ``=` `1` `                    ``l ``+``=` `1` ` `  `            ``elif` `(count_on): ` `                ``Sum` `+``=` `((mul ``*` `l ``*` `(l ``+` `1``)) ``/``/` `2``) ` `                ``count_on ``=` `0` `                ``l ``=` `0` `             `  `        ``if` `(count_on):  ` `            ``Sum` `+``=` `((mul ``*` `l ``*` `(l ``+` `1``)) ``/``/` `2``) ` `            ``count_on ``=` `0` `            ``l ``=` `0` `         `  `        ``# updating the multiplier ` `        ``mul ``*``=` `2` `     `  `    ``# returning the Sum ` `    ``return` `Sum` ` `  `# Driver Code ` `arr ``=` `[``7``, ``1``, ``1``, ``5``] ` ` `  `n ``=` `len``(arr) ` ` `  `print``(findAndSum(arr, n)) ` ` `  `# This code is contributed by Mohit Kumar `

## C#

 `// C# program to find sum of bitwise AND ` `// of all subarrays ` `using` `System; ` ` `  `class` `GFG ` `{ ` ` `  `// Function to find the sum of ` `// bitwise AND of all subarrays ` `static` `int` `findAndSum(``int` `[]arr, ``int` `n) ` `{ ` `    ``// variable to store ` `    ``// the final sum ` `    ``int` `sum = 0; ` ` `  `    ``// multiplier ` `    ``int` `mul = 1; ` ` `  `    ``for` `(``int` `i = 0; i < 30; i++)  ` `    ``{ ` `        ``// variable to check if ` `        ``// counting is on ` `        ``bool` `count_on = ``false``; ` ` `  `        ``// variable to store the ` `        ``// length of the subarrays ` `        ``int` `l = 0; ` ` `  `        ``// loop to find the contiguous ` `        ``// segments ` `        ``for` `(``int` `j = 0; j < n; j++)  ` `        ``{ ` `            ``if` `((arr[j] & (1 << i)) > 0) ` `                ``if` `(count_on) ` `                    ``l++; ` `                ``else` `                ``{ ` `                    ``count_on = ``true``; ` `                    ``l++; ` `                ``} ` ` `  `            ``else` `if` `(count_on) ` `            ``{ ` `                ``sum += ((mul * l * (l + 1)) / 2); ` `                ``count_on = ``false``; ` `                ``l = 0; ` `            ``} ` `        ``} ` ` `  `        ``if` `(count_on)  ` `        ``{ ` `            ``sum += ((mul * l * (l + 1)) / 2); ` `            ``count_on = ``false``; ` `            ``l = 0; ` `        ``} ` ` `  `        ``// updating the multiplier ` `        ``mul *= 2; ` `    ``} ` ` `  `    ``// returning the sum ` `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main() ` `{ ` `    ``int` `[]arr = { 7, 1, 1, 5 }; ` `    ``int` `n = arr.Length; ` ` `  `    ``Console.Write(findAndSum(arr, n)); ` `} ` `} ` ` `  `// This code is contributed  ` `// by Akanksha Rai `

## PHP

 ` 0)  ` `                ``if` `(``\$count_on``)  ` `                    ``\$l``++;  ` `                ``else`  `                ``{  ` `                    ``\$count_on` `= 1;  ` `                    ``\$l``++;  ` `                ``}  ` ` `  `            ``else` `if` `(``\$count_on``)  ` `            ``{  ` `                ``\$sum` `+= ((``\$mul` `* ``\$l` `* (``\$l` `+ 1)) / 2);  ` `                ``\$count_on` `= 0;  ` `                ``\$l` `= 0;  ` `            ``}  ` `        ``}  ` ` `  `        ``if` `(``\$count_on``)  ` `        ``{  ` `            ``\$sum` `+= ((``\$mul` `* ``\$l` `* (``\$l` `+ 1)) / 2);  ` `            ``\$count_on` `= 0;  ` `            ``\$l` `= 0;  ` `        ``}  ` ` `  `        ``// updating the multiplier  ` `        ``\$mul` `*= 2;  ` `    ``}  ` ` `  `    ``// returning the sum  ` `    ``return` `\$sum``;  ` `}  ` ` `  `// Driver Code  ` `\$arr` `= ``array``( 7, 1, 1, 5 );  ` ` `  `\$n` `= sizeof(``\$arr``);  ` ` `  `echo` `findAndSum(``\$arr``, ``\$n``);  ` ` `  `// This code is contributed by Ryuga ` `?> `

Output:

```20
```

Time Complexity: O(N*log(max(A))

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