Problem Description
Given a 0-indexed n x n integer matrix grid, return the number of pairs (r_i, c_j) such that row r_i and column c_j are equal. A row and column pair is considered equal if they contain the same elements in the same order (i.e., an equal array).
Key Insights
- Rows and columns must be compared element-wise for equality.
- Using a hash table can efficiently store and count occurrences of row representations.
- The number of equal row-column pairs can be derived from the counts of matching rows and columns stored in the hash table.
Space and Time Complexity
Time Complexity: O(n^2) - We need to iterate through each element of the n x n matrix to build row and column representations.
Space Complexity: O(n) - We use a hash table to store at most n row representations.
Solution
To solve the problem, we utilize a hash table (or dictionary) to keep track of the frequency of each row representation in the matrix. Each row is converted to a tuple (or a string) to allow it to be stored as a key in the hash table. After processing all rows, we then iterate over each column, checking if its representation exists in the hash table. The count of equal row-column pairs is accumulated based on the frequency stored in the hash table.