Find the number of corner rectangles that can be formed from given Matrix
Given a binary matrix mat[][] of dimensions N*M, the task is to find the number of corner rectangles that can be formed. A corner rectangle is defined as the submatrix having 1s on the corners of it and each 1s must belong to a unique cell in that submatrix.
Examples:
Input: mat[][] = {{1, 0, 1}, {0, 0, 0}, {1, 0, 1}}
Output: 1
Explanation:
There exists only one submatrix satisfying the given formula represented by bold as:
1 0 1
0 0 0
1 0 1Input: mat[][] = {{1, 1, 1}, {1, 1, 1}, {1, 1, 1}}
Output: 9
Approach: The given problem can be solved by using the concepts of all distinct possible pairs from N points which are given by NC2. The idea is to store the frequency of pair of cells (i, j) having the values as 1s in the map of pairs, say M. After generating the frequency map find the total count of corners rectangle formed using the above formula. Follow the steps below to solve the given problem:
- Initialize a variable, say count that stores the resultant count of corner rectangle.
- Initialize a map, say m[] that stores the frequency of the cell (i, j) having values as 1.
- Iterate over the range [0, M) using the variable i and perform the following tasks:
- Iterate over the range [0, N) using the variable j and if mat[i][j] equals 1 then Iterate over the range [j_+1, N) using the variable k and if the value of mat[i][k] equals 1 then increase the count of m[{j, k}] by 1.
- Traverse over the map m[] using the variable it and add the value of it.second*(it.second – 1)/2 to the variable count.
- After performing the above steps, print the value of count as the answer.
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include <bits/stdc++.h> using namespace std; // Function to find all possible rectangles // having distinct corners as 1s int countCornerRectangles( vector<vector<int> >& mat) { // Stores the count of rectangles int count = 0; int N = mat.size(); int M = mat[0].size(); // Map to store the frequency map<pair<int, int>, int> m; // Iterate over each row for (int i = 0; i < N; i++) { // Iterate over each cell for (int j = 0; j < M; j++) { if (mat[i][j] == 1) { // Check for 1's of th // same column pair for (int k = j + 1; k < M; k++) { if (mat[i][k] == 1) { m[{ j, k }]++; } } } } } // For any pair of column cells (x, y) // If the frequency is n. Then choosing // any 2 will form a rectangle for (auto& it : m) { count += (it.second * (it.second - 1)) / 2; } return count; } // Driver Code int main() { vector<vector<int> > mat = { { 1, 1, 1 }, { 1, 1, 1 }, { 1, 1, 1 } }; cout << countCornerRectangles(mat); return 0; } |
9
Time Complexity: O(N*M2)
Auxiliary Space: O(M2)
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