# Sum of bitwise AND of all submatrices

Given a NxN matrix, the task is to find the sum of bit-wise AND of all of its rectangular sub-matrices.

Examples:

```Input : arr[][] = {{1, 1, 1},
{1, 1, 1},
{1, 1, 1}}
Output : 36
Explanation: All the possible submatrices will have AND value 1.
Since, there are 36 submatrices in total, ans = 36

Input : arr[][] = {{9, 7, 4},
{8, 9, 2},
{11, 11, 5}}
Output : 135
```

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

Naive Solution: A simple solution is to generate all of the sub-matrices and find the required AND for each of them. The time complexity of this approach will be O(N6).

Efficient Approach: 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-matrices with bit-wise and(&)) with ith bit set. Let us suppose, there are ‘Si‘ number of sub-matrices with ith bit set. For, ith bit, sum can be updated as sum += (2i * Si).

For each bit ‘i’, create a boolean matrix set_bit which stores ‘1’ at an index (R, C) if ith bit of arr[R][C] is set. Otherwise, it stores ‘0’. Then, for this boolean array, we try to find the number of rectangular submatrices with all 1s(Si). For, ith bit, the final sum will be updated as:

```sum += 2i * Si
```

Below is the implementation of the above approach:

## C++

 `// C++ program to find sum of Bit-wise AND ` `// of all submatrices ` ` `  `#include ` `#include ` ` `  `using` `namespace` `std; ` ` `  `#define n 3 ` ` `  `// Function to find prefix-count for each row ` `// from right to left ` `void` `findPrefixCount(``int` `p_arr[][n], ``bool` `set_bit[][n]) ` `{ ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``for` `(``int` `j = n - 1; j >= 0; j--) { ` `            ``if` `(!set_bit[i][j]) ` `                ``continue``; ` ` `  `            ``if` `(j != n - 1) ` `                ``p_arr[i][j] += p_arr[i][j + 1]; ` ` `  `            ``p_arr[i][j] += (``int``)set_bit[i][j]; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to find the number of submatrices ` `// with all 1s ` `int` `matrixAllOne(``bool` `set_bit[][n]) ` `{ ` `    ``// Array to store required prefix count of 1s from ` `    ``// right to left for boolean array ` `    ``int` `p_arr[n][n] = { 0 }; ` ` `  `    ``findPrefixCount(p_arr, set_bit); ` ` `  `    ``// Variable to store the final answer ` `    ``int` `ans = 0; ` ` `  `    ``// For each index of a column, determine the number ` `    ``// of sub-matrices starting from that index ` `    ``// and has all 1s ` `    ``for` `(``int` `j = 0; j < n; j++) { ` `        ``int` `i = n - 1; ` ` `  `        ``// Stack to store elements and the count ` `        ``// of the numbers they popped ` `        ``// First part of pair is value of inserted element ` `        ``// Second part is count of the number of elements ` `        ``// pushed before with a greater value ` `        ``stack > q; ` ` `  `        ``// variable to store the number of submatrices ` `        ``// with all 1s ` `        ``int` `to_sum = 0; ` ` `  `        ``while` `(i >= 0) { ` `            ``int` `c = 0; ` `            ``while` `(q.size() != 0 and q.top().first > p_arr[i][j]) { ` `                ``to_sum -= (q.top().second + 1) * (q.top().first - p_arr[i][j]); ` `                ``c += q.top().second + 1; ` `                ``q.pop(); ` `            ``} ` ` `  `            ``to_sum += p_arr[i][j]; ` `            ``ans += to_sum; ` ` `  `            ``q.push({ p_arr[i][j], c }); ` `            ``i--; ` `        ``} ` `    ``} ` ` `  `    ``return` `ans; ` `} ` ` `  `// Function to find the sum of Bitwise-AND ` `// of all submatrices ` `int` `sumAndMatrix(``int` `arr[][n]) ` `{ ` `    ``int` `sum = 0; ` ` `  `    ``int` `mul = 1; ` ` `  `    ``for` `(``int` `i = 0; i < 30; i++) { ` `        ``// matrix to store the status ` `        ``// of ith bit of each element ` `        ``// of matrix arr ` `        ``bool` `set_bit[n][n]; ` ` `  `        ``for` `(``int` `R = 0; R < n; R++) ` `            ``for` `(``int` `C = 0; C < n; C++) ` `                ``set_bit[R][C] = ((arr[R][C] & (1 << i)) != 0); ` ` `  `        ``sum += (mul * matrixAllOne(set_bit)); ` ` `  `        ``mul *= 2; ` `    ``} ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `arr[][n] = { { 9, 7, 4 }, ` `                     ``{ 8, 9, 2 }, ` `                     ``{ 11, 11, 5 } }; ` ` `  `    ``cout << sumAndMatrix(arr); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to find sum of Bit-wise AND  ` `// of all submatrices  ` `import` `java.util.*; ` ` `  `class` `GFG ` `{ ` ` `  `static` `int` `n = ``3``;  ` ` `  `// Function to find prefix-count for   ` `// each row from right to left  ` `static` `void` `findPrefixCount(``int` `p_arr[][],  ` `                            ``boolean` `set_bit[][])  ` `{  ` `    ``for` `(``int` `i = ``0``; i < n; i++)  ` `    ``{  ` `        ``for` `(``int` `j = n - ``1``; j >= ``0``; j--)  ` `        ``{  ` `            ``if` `(!set_bit[i][j])  ` `                ``continue``;  ` ` `  `            ``if` `(j != n - ``1``)  ` `                ``p_arr[i][j] += p_arr[i][j + ``1``];  ` ` `  `            ``p_arr[i][j] += (set_bit[i][j]) ? ``1` `: ``0``;  ` `        ``}  ` `    ``}  ` `}  ` `static` `class` `pair ` `{ ` `    ``int` `first,second; ` `    ``pair(){} ` `     `  `    ``pair(``int` `a, ``int` `b) ` `    ``{ ` `        ``first = a; ` `        ``second = b; ` `    ``} ` `} ` ` `  `// Function to find the number of  ` `// submatrices with all 1s  ` `static` `int` `matrixAllOne(``boolean` `set_bit[][])  ` `{  ` `    ``// Array to store required prefix count of 1s from  ` `    ``// right to left for boolean array  ` `    ``int` `p_arr[][] = ``new` `int``[n][n];  ` `     `  `    ``for``(``int` `i = ``0``; i < n; i++) ` `        ``for``(``int` `j = ``0``; j < n;j++) ` `            ``p_arr[i][j] = ``0``; ` `     `  `    ``findPrefixCount(p_arr, set_bit);  ` ` `  `    ``// Variable to store the final answer  ` `    ``int` `ans = ``0``;  ` ` `  `    ``// For each index of a column, determine the number  ` `    ``// of sub-matrices starting from that index  ` `    ``// and has all 1s  ` `    ``for` `(``int` `j = ``0``; j < n; j++)  ` `    ``{  ` `        ``int` `i = n - ``1``;  ` ` `  `        ``// Stack to store elements and the count  ` `        ``// of the numbers they popped  ` `        ``// First part of pair is value of inserted element  ` `        ``// Second part is count of the number of elements  ` `        ``// pushed before with a greater value  ` `        ``Stack q = ``new` `Stack();  ` ` `  `        ``// variable to store the number of submatrices  ` `        ``// with all 1s  ` `        ``int` `to_sum = ``0``;  ` ` `  `        ``while` `(i >= ``0``) ` `        ``{  ` `            ``int` `c = ``0``;  ` `            ``while` `(q.size() != ``0` `&&  ` `                    ``q.peek().first > p_arr[i][j])  ` `            ``{  ` `                ``to_sum -= (q.peek().second + ``1``) * ` `                            ``(q.peek().first - p_arr[i][j]);  ` `                ``c += q.peek().second + ``1``;  ` `                ``q.pop();  ` `            ``}  ` ` `  `            ``to_sum += p_arr[i][j];  ` `            ``ans += to_sum;  ` ` `  `            ``q.push(``new` `pair( p_arr[i][j], c ));  ` `            ``i--;  ` `        ``}  ` `    ``}  ` `    ``return` `ans;  ` `}  ` ` `  `// Function to find sum of Bitwise-OR of  ` `// all submatrices  ` `static` `int` `sumAndMatrix(``int` `arr[][])  ` `{  ` `    ``int` `sum = ``0``;  ` ` `  `    ``int` `mul = ``1``;  ` ` `  `    ``for` `(``int` `i = ``0``; i < ``30``; i++)  ` `    ``{  ` `        ``// matrix to store the status  ` `        ``// of ith bit of each element  ` `        ``// of matrix arr  ` `        ``boolean` `set_bit[][] = ``new` `boolean``[n][n];  ` ` `  `        ``for` `(``int` `R = ``0``; R < n; R++)  ` `            ``for` `(``int` `C = ``0``; C < n; C++)  ` `                ``set_bit[R][C] = ((arr[R][C] & (``1` `<< i)) != ``0``);  ` ` `  `        ``sum += (mul * matrixAllOne(set_bit));  ` ` `  `        ``mul *= ``2``;  ` `    ``}  ` `    ``return` `sum;  ` `}  ` ` `  `// Driver Code  ` `public` `static` `void` `main(String args[]) ` `{  ` `    ``int` `arr[][] = { { ``9``, ``7``, ``4` `},  ` `                    ``{ ``8``, ``9``, ``2` `},  ` `                    ``{ ``11``, ``11``, ``5` `} };  ` ` `  `    ``System.out.println( sumAndMatrix(arr));  ` `}  ` `} ` ` `  `// This code is contributed by Arnab Kundu  `

## Python3

 `# Python3 program to find sum of  ` `# Bitwise-AND of all submatrices  ` ` `  `# Function to find prefix-count for  ` `# each row from right to left  ` `def` `findPrefixCount(p_arr, set_bit):  ` ` `  `    ``for` `i ``in` `range``(``0``, n): ` `        ``for` `j ``in` `range``(n ``-` `1``, ``-``1``, ``-``1``):  ` ` `  `            ``if` `not` `set_bit[i][j]:  ` `                ``continue` `            ``if` `j !``=` `n ``-` `1``:  ` `                ``p_arr[i][j] ``+``=` `p_arr[i][j ``+` `1``]  ` ` `  `            ``p_arr[i][j] ``+``=` `int``(set_bit[i][j])  ` ` `  `# Function to create a boolean matrix  ` `# set_bit which stores ‘1’ at an index  ` `# (R, C) if ith bit of arr[R][C] is set.  ` `def` `matrixAllOne(set_bit):  ` ` `  `    ``# Array to store prefix count of zeros  ` `    ``# from right to left for boolean array  ` `    ``p_arr ``=` `[[``0` `for` `i ``in` `range``(n)]  ` `                ``for` `j ``in` `range``(n)]  ` ` `  `    ``findPrefixCount(p_arr, set_bit)  ` ` `  `    ``# Variable to store the final answer  ` `    ``ans ``=` `0` ` `  `    ``# For each index of a column we  ` `    ``# will try to determine the number  ` `    ``# of sub-matrices starting from  ` `    ``# that index and has all 1s  ` `    ``for` `j ``in` `range``(``0``, n):  ` ` `  `        ``i ``=` `n ``-` `1` `         `  `        ``# stack to store elements and the  ` `        ``# count of the numbers they popped  ` ` `  `        ``# First part of pair will be the  ` `        ``# value of inserted element.  ` `        ``# Second part will be the count  ` `        ``# of the number of elements pushed  ` `        ``# before with a greater value  ` `        ``q ``=` `[]  ` ` `  `        ``# Variable to store the number  ` `        ``# of submatrices with all 0s  ` `        ``to_sum ``=` `0` `         `  `        ``while` `i >``=` `0``:  ` ` `  `            ``c ``=` `0` `            ``while` `(``len``(q) !``=` `0` `and` `                   ``q[``-``1``][``0``] > p_arr[i][j]):  ` ` `  `                ``to_sum ``-``=` `((q[``-``1``][``1``] ``+` `1``) ``*`  `                           ``(q[``-``1``][``0``] ``-` `p_arr[i][j]))  ` ` `  `                ``c ``+``=` `q.pop()[``1``] ``+` `1` ` `  `            ``to_sum ``+``=` `p_arr[i][j]  ` `            ``ans ``+``=` `to_sum  ` ` `  `            ``q.append((p_arr[i][j], c))  ` `            ``i ``-``=` `1` ` `  `    ``# Return the final answer  ` `    ``return` `ans  ` ` `  `# Function to find sum of  ` `# Bitwise-AND of all submatrices  ` `def` `sumAndMatrix(arr):  ` ` `  `    ``Sum``, mul ``=` `0``, ``1` `    ``for` `i ``in` `range``(``0``, ``30``):  ` `     `  `        ``# matrix to store the status  ` `        ``# of ith bit of each element  ` `        ``# of matrix arr  ` `        ``set_bit ``=` `[[``False` `for` `i ``in` `range``(n)]  ` `                          ``for` `j ``in` `range``(n)] ` ` `  `        ``for` `R ``in` `range``(``0``, n):  ` `            ``for` `C ``in` `range``(``0``, n):  ` `                ``set_bit[R][C] ``=` `((arr[R][C] &  ` `                                 ``(``1` `<< i)) !``=` `0``)  ` ` `  `        ``Sum` `+``=` `(mul ``*` `matrixAllOne(set_bit))  ` `        ``mul ``*``=` `2` ` `  `    ``return` `Sum` ` `  `# Driver Code  ` `if` `__name__ ``=``=` `"__main__"``: ` `     `  `    ``n ``=` `3` `    ``arr ``=` `[[``9``, ``7``, ``4``],  ` `        ``[``8``, ``9``, ``2``],  ` `        ``[``11``, ``11``, ``5``]]  ` ` `  `    ``print``(sumAndMatrix(arr))  ` ` `  `# This code is contributed by Rituraj Jain `

## C#

 `// C# program to find sum of Bit-wise AND  ` `// of all submatrices ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG ` `{ ` ` `  `static` `int` `n = 3;  ` ` `  `// Function to find prefix-count for  ` `// each row from right to left  ` `static` `void` `findPrefixCount(``int` `[,]p_arr,  ` `                            ``bool` `[,]set_bit)  ` `{  ` `    ``for` `(``int` `i = 0; i < n; i++)  ` `    ``{  ` `        ``for` `(``int` `j = n - 1; j >= 0; j--)  ` `        ``{  ` `            ``if` `(!set_bit[i, j])  ` `                ``continue``;  ` ` `  `            ``if` `(j != n - 1)  ` `                ``p_arr[i, j] += p_arr[i, j + 1];  ` ` `  `            ``p_arr[i, j] += (set_bit[i, j]) ? 1 : 0;  ` `        ``}  ` `    ``}  ` `}  ` `public` `class` `pair ` `{ ` `    ``public` `int` `first,second; ` `    ``public` `pair(){} ` `     `  `    ``public` `pair(``int` `a, ``int` `b) ` `    ``{ ` `        ``first = a; ` `        ``second = b; ` `    ``} ` `} ` ` `  `// Function to find the number of  ` `// submatrices with all 1s  ` `static` `int` `matrixAllOne(``bool` `[,]set_bit)  ` `{  ` `    ``// Array to store required prefix count of 1s from  ` `    ``// right to left for boolean array  ` `    ``int` `[,]p_arr = ``new` `int``[n, n];  ` `     `  `    ``for``(``int` `i = 0; i < n; i++) ` `        ``for``(``int` `j = 0; j < n;j++) ` `            ``p_arr[i, j] = 0; ` `     `  `    ``findPrefixCount(p_arr, set_bit);  ` ` `  `    ``// Variable to store the final answer  ` `    ``int` `ans = 0;  ` ` `  `    ``// For each index of a column, determine the number  ` `    ``// of sub-matrices starting from that index  ` `    ``// and has all 1s  ` `    ``for` `(``int` `j = 0; j < n; j++)  ` `    ``{  ` `        ``int` `i = n - 1;  ` ` `  `        ``// Stack to store elements and the count  ` `        ``// of the numbers they popped  ` `        ``// First part of pair is value of inserted element  ` `        ``// Second part is count of the number of elements  ` `        ``// pushed before with a greater value  ` `        ``Stack q = ``new` `Stack();  ` ` `  `        ``// variable to store the number of submatrices  ` `        ``// with all 1s  ` `        ``int` `to_sum = 0;  ` ` `  `        ``while` `(i >= 0) ` `        ``{  ` `            ``int` `c = 0;  ` `            ``while` `(q.Count != 0 &&  ` `                    ``q.Peek().first > p_arr[i,j])  ` `            ``{  ` `                ``to_sum -= (q.Peek().second + 1) * ` `                            ``(q.Peek().first - p_arr[i,j]);  ` `                ``c += q.Peek().second + 1;  ` `                ``q.Pop();  ` `            ``}  ` ` `  `            ``to_sum += p_arr[i,j];  ` `            ``ans += to_sum;  ` ` `  `            ``q.Push(``new` `pair( p_arr[i,j], c ));  ` `            ``i--;  ` `        ``}  ` `    ``}  ` `    ``return` `ans;  ` `}  ` ` `  `// Function to find sum of Bitwise-OR of  ` `// all submatrices  ` `static` `int` `sumAndMatrix(``int` `[,]arr)  ` `{  ` `    ``int` `sum = 0;  ` ` `  `    ``int` `mul = 1;  ` ` `  `    ``for` `(``int` `i = 0; i < 30; i++)  ` `    ``{  ` `        ``// matrix to store the status  ` `        ``// of ith bit of each element  ` `        ``// of matrix arr  ` `        ``bool` `[,]set_bit = ``new` `bool``[n,n];  ` ` `  `        ``for` `(``int` `R = 0; R < n; R++)  ` `            ``for` `(``int` `C = 0; C < n; C++)  ` `                ``set_bit[R, C] = ((arr[R, C] & (1 << i)) != 0);  ` ` `  `        ``sum += (mul * matrixAllOne(set_bit));  ` ` `  `        ``mul *= 2;  ` `    ``}  ` `    ``return` `sum;  ` `}  ` ` `  `// Driver Code  ` `public` `static` `void` `Main(String []args) ` `{  ` `    ``int` `[,]arr = { { 9, 7, 4 },  ` `                    ``{ 8, 9, 2 },  ` `                    ``{ 11, 11, 5 } };  ` ` `  `    ``Console.WriteLine(sumAndMatrix(arr));  ` `}  ` `} ` ` `  `// This code contributed by Rajput-Ji `

Output:

```135
```

Time Complexity: O(N2).

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