What are Equal Row and Column Pairs?

Equal row and column pairs is a common topic in developer interviews that tests a candidate's problem-solving skills. This concept usually relates to multi-dimensional arrays, often focusing on counting the pairs of rows and columns where the sums of the elements are equal.

Problem Description

Given a 2D array (or matrix), the goal is to find the number of pairs consisting of a row and a column such that the sum of the elements in that row equals the sum of the elements in that column. This problem helps assess a candidate's ability to work with arrays, loop through data structures, and optimize solutions.

The basic premise involves calculating the sums of each row and each column and then determining how many combinations of those sums match.

Example

Let's consider an example matrix:

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1 2 3
4 5 6
7 8 9

The sum of the rows would be:

• Row 1: 1 + 2 + 3 = 6
• Row 2: 4 + 5 + 6 = 15
• Row 3: 7 + 8 + 9 = 24

The sum of the columns would be:

• Column 1: 1 + 4 + 7 = 12
• Column 2: 2 + 5 + 8 = 15
• Column 3: 3 + 6 + 9 = 18

In this case, looking for equal row and column pairs:

• The second row sum (15) equals the second column sum (15).

Thus, we have one pair: (Row 2, Column 2).

Common Interview Questions

Question 1: How would you approach this problem?

Example Answer: To solve this, I would:

1. Calculate the sums of all rows and store them in an array.
2. Calculate the sums of all columns and store them in another array.
3. Use a hash map to count the frequency of each row sum.
4. For each column sum, check if it exists in the hash map and sum the frequency counts.

Question 2: How can you optimize your solution?

Example Answer: The time complexity of the naive approach is O(n * m) for calculating row and column sums, followed by O(n + m) for checking pairs. However, by using a hash map to store row sums, we can reduce our lookups to O(1). Thus, the overall expected time complexity becomes O(n + m), where n is the number of rows and m is the number of columns. This is efficient and allows for quick pair lookups.

Question 3: Can you write a code snippet for this?

Example Answer: Sure, here’s a Python implementation:

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def equal_row_column_pairs(matrix):
    if not matrix:
        return 0
    
    row_sums = {}
    count = 0

    # Calculate row sums
    for row in matrix:
        row_sum = sum(row)
        if row_sum in row_sums:
            row_sums[row_sum] += 1
        else:
            row_sums[row_sum] = 1

    # Calculate column sums and count pairs
    num_cols = len(matrix)
    for col in range(num_cols):
        col_sum = sum(matrix[row][col] for row in range(len(matrix)))
        if col_sum in row_sums:
            count += row_sums[col_sum]
    
    return count

# Example usage:
matrix = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]
print(equal_row_column_pairs(matrix))  # Output: 1

This code defines a function that first computes row sums and stores them in a dictionary. Then, it calculates the column sums and checks corresponding pairs in the dictionary to count how many rows and columns have equal sums.

Being able to reason through problems like equal row and column pairs shows analytical skills and programming proficiency. Approaching such questions methodically ensures clarity and efficiency, making a candidate stand out in interviews.