Prerequisites: Matrix, Map/HashMap.
Problem: Given a N x N integer matrix, find the no. of pairs of row and column which are equal.
Solution Approach: The main idea of the solution is to count the number of pairs of rows and columns in a given integer matrix that are equal. This is achieved by constructing a map where each key-value pair represents a unique row vector and its frequency in the matrix. Initially, the map is empty. We then iterate through each row of the matrix and insert its vector representation into the map, incrementing the frequency count if the row vector already exists in the map. After constructing the map, we iterate through each column of the matrix. For each column, we construct a vector containing its elements and check the map to find the frequency of this column vector. The frequency represents the number of rows in the matrix that have the same elements in the corresponding column. By summing up these frequencies for each column, we obtain the total count of pairs of equal rows and columns. This approach efficiently handles the problem by leveraging the map data structure to store and manipulate row vectors, resulting in a time complexity of O(N * log N), where N is the size of the matrix.
Time Complexity: O(N*logN + N * M) = O(N * M)
Space Complexity: (N * M), because we’re storing almost entire matrix in map in worst case.