Number of submatrices with OR value 1

Given a N*N binary matrix, task is to find the count of rectangular sub-matrices with OR value 1.

Examples:

Input : arr[][] = {{0, 0, 0},
                   {0, 0, 0},
                   {0, 0, 0}}
Output : 0
Explanation: All the submatrices will have an OR value 0.
Thus, ans = 0.

Input : arr[][] = {{0, 0, 0},
                  {0, 1, 0},
                  {0, 0, 0}}
Output : 16

A Simple Solution will be to generate all the possible sub-matrices and then check if any of the values inside them are 1. If for a sub-matrix, atleast a single element is one, we increment the value of final answer for one. The time complexity of above approach is O(n6).



Better approach: Lets have a look at this problem in an other way. We will now try to find the number of submatrices with all 0s. And for the final answer, we will substract this value from the total number of submatrices.

To optimize the process, for every index of the matrix, we will try to find the number of submatrices starting from that index having all 0s in it.

Our first step towards solving this problem is creating a matrix ‘p_arr’.

  • For each index (R, C), if arr[R][C] equals 0, then in p_arr[R][C], we will store the number of 0s to the right of the cell(R, C) along row ‘R’ before we encounter ‘1’ or end of the array plus one.
  • If arr[R][C] equals 1, the p_arr[R][C] equals zero.

For creating this matrix, we will use the following recurrence relation.

IF arr[R][C] is 0
    p_arr[R][C] = p_arr[R][C+1] + 1
ELSE 
    p_arr[R][C] = 0

arr[][] = {{1, 0, 0, 0},
           {0, 1, 0, 1},
           {0, 1, 0, 0},
           {0, 0, 0, 0}}

p_arr[][] for above will look like
          {{0, 3, 2, 1},
           {1, 0, 1, 0},
           {1, 0, 2, 1},
           {4, 3, 2, 1}}

Once, we have the required matrix p_arr, we will start processing the matrix ‘p_arr’ column-wise. If we are processing jth column of the matrix ‘p_arr’, then for each element ‘i’ of this column, we will try to find the number of sub-matrices starting from cell (i, j) with all 0s.

For this, we can use stack data structure.

Algorithm:

  1. Initialize a stack ‘q’ to store the value of the elements getting pushed along with the count(Cij) of the number of elements that were pushed in the stack with a value strictly greater than the value of the current element. We will use pair to tie up the two data together.
    Initialize a variable to_sum with 0. At each step, this variable is updated to store the number of submatrices with all 0s starting from the element being pushed at that step. Thus, using ‘to_sum’, we update the count of number of submatrices with all 0s at each step.
  2. For a column ‘j’, at any step ‘i’, we will prepare to push p_arr[i][j] in the stack. Let Qt represent the topmost element of the stack and Ct represent the number of elements previously pushed in the stack with a value greater than the top-most element of the stack. Before pushing an element ‘p_arr[i][j]’ in the stack, while the stack is not empty or topmost element is greater than the number to be pushed, keep popping the topmost element of the stack and at the same time update to_sum as to_sum += (Ct + 1) * (Qt – p_arr[i][j]). Let Ci, jrepresent the number of elements greater than the current element that were pushed in this stack previously. We also need to keep a track of Ci, j. Thus, before popping an element, we update Ci, j as Ci, j += Ct along with to_sum.
  3. We update the number of submatrices with all zeros as count_zero_submatrices += to_sum.
  4. Finally, we push that element in the stack after pairing it with Ci, j.

Total number of sub-matrices in a N*N matrix equals:

(N2 * (N + 1)2)/4

Thus, the final answer will be:

ans = (N2 * (N + 1)2)/4 - count_zero_submatrices 

We create the prefix-array in O(N2) and for each column we push an element in the stack or pop it out only once. Thus, the time complexity of this algorithm is O(N2).

Below is the implementation of the above approach:

C++

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// C++ program to count number of submatrices
// with OR value 1
  
#include <iostream>
#include <stack>
#define n 3
using namespace std;
  
// Function to find required prefix-count for each row
// from right to left
void findPrefixCount(int p_arr[][n], bool arr[][n])
{
    for (int i = 0; i < n; i++)
        for (int j = n - 1; j >= 0; j--) {
  
            if (arr[i][j])
                continue;
            if (j != n - 1)
                p_arr[i][j] += p_arr[i][j + 1];
  
            p_arr[i][j] += (int)(!arr[i][j]);
        }
}
  
// Function to find the count of submatrices
// with OR value 1
int matrixOrValueOne(bool arr[][n])
{
    // Array to store prefix count of zeros from
    // right to left for boolean array
    int p_arr[n][n] = { 0 };
  
    findPrefixCount(p_arr, arr);
  
    // Variable to store the count of
    // submatrices with OR value 0
    int count_zero_submatrices = 0;
  
    // Loop to evaluate each column of
    // the prefix matrix uniquely.
    // 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 (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 will be the
        // value of inserted element.
        // Second part will be the count
        // of the number of elements pushed
        // before with a greater value
        stack<pair<int, int> > q;
  
        // Variable to store the number of submatrices
        // with all 0s
        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];
  
            count_zero_submatrices += to_sum;
  
            q.push({ p_arr[i][j], c });
  
            i--;
        }
    }
  
    // Return the final answer
    return (n * (n + 1) * n * (n + 1)) / 4
           - count_zero_submatrices;
}
  
// Driver Code
int main()
{
    bool arr[][n] = { { 0, 0, 0 },
                      { 0, 1, 0 },
                      { 0, 0, 0 } };
  
    cout << matrixOrValueOne(arr);
  
    return 0;
}

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Python3

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# Python3 program to count number 
# of submatrices with OR value 1 
  
# Function to find required prefix-count 
# for each row from right to left 
def findPrefixCount(p_arr, arr): 
  
    for i in range(0, n):
        for j in range(n - 1, -1, -1): 
  
            if arr[i][j]: 
                continue
            if j != n - 1
                p_arr[i][j] += p_arr[i][j + 1
  
            p_arr[i][j] += int(not arr[i][j]) 
  
# Function to find the count 
# of submatrices with OR value 1 
def matrixOrValueOne(arr): 
  
    # 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, arr) 
  
    # Variable to store the count of 
    # submatrices with OR value 0 
    count_zero_submatrices = 0
  
    # Loop to evaluate each column of 
    # the prefix matrix uniquely. 
    # 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] 
            count_zero_submatrices += to_sum 
  
            q.append((p_arr[i][j], c)) 
            i -= 1
  
    # Return the final answer 
    return ((n * (n + 1) * n * (n + 1)) // 
             4 - count_zero_submatrices) 
  
# Driver Code 
if __name__ == "__main__"
  
    n = 3
    arr = [[0, 0, 0], 
           [0, 1, 0], 
           [0, 0, 0]] 
  
    print(matrixOrValueOne(arr))
  
# This code is contributed by Rituraj Jain

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Output:

16


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