Matrix Diagonal Sum

Identifying Problem Isomorphism

“Matrix Diagonal Sum” can be approximately mapped to “Toeplitz Matrix”.

Reasoning:

Both involve the traversal of a 2D matrix in a diagonal manner and the examination of those diagonal elements.

1. In “Matrix Diagonal Sum”, you’re required to calculate the sum of elements on the diagonals of a square matrix. This problem is simple in terms of understanding the requirements and implementing a solution. It mainly involves the traversal of the matrix diagonally and summing up those elements.

2. “Toeplitz Matrix”, while more complex, requires you to traverse the matrix in a similar diagonal manner. However, instead of summing elements, you’re checking if each diagonal from top-left to bottom-right has the same elements.

These are approximate isomorphs. They are related as they both necessitate a similar diagonal traversal of a matrix, although the specific operations differ. Thus, if you’re comfortable with one, the other problem should also be approachable.

The problem requires us to calculate the sum of elements along the primary and secondary diagonals of a square matrix.

For a square matrix, the primary diagonal is the set of elements mat[i][i] for all i, where mat is the given matrix.

The secondary diagonal is the set of elements mat[i][n-i-1] for all i, where n is the size of the matrix.

If the matrix size n is odd, the middle element belongs to both diagonals, so we need to avoid double-counting it.

The algorithm for this problem follows these steps:

1. Initialize a variable total to hold the sum, and set it to 0.
2. Iterate over the matrix from i = 0 to n-1.
3. For each i, add mat[i][i] and mat[i][n-i-1] to total.
4. If the size of the matrix n is odd, subtract the middle element mat[n//2][n//2] from total to correct for double-counting.
5. Return total.

This can be implemented in Python as follows:

1
2
3
4
5
6
7
8
9

class Solution:
    def diagonalSum(self, mat: List[List[int]]) -> int:
        n = len(mat)
        total = 0
        for i in range(n):
            total += mat[i][i] + mat[i][n-i-1]
        if n % 2 == 1:
            total -= mat[n//2][n//2]
        return total

This solution works by iterating over each row of the matrix, summing the elements on the primary and secondary diagonals, and correcting for the double-counting of the middle element if the matrix size is odd. It has a time complexity of O(n), as we perform a constant amount of work for each row of the matrix.

Let’s use a 3x3 matrix as an example:

mat = [[a, b, c],
       [d, e, f],
       [g, h, i]]

The primary diagonal consists of elements a, e, i. The secondary diagonal consists of elements c, e, g. The indices for these elements in matrix mat are as follows:

From the table, you can observe that for the primary diagonal, the row index and column index are always the same (i.e., i = j). For the secondary diagonal, the row index and column index always sum up to the size of the matrix minus 1 (i.e., i + j = n - 1, where n is the size of the matrix).

Using these observations, we can traverse the matrix as follows:

1. Start a loop with i from 0 to n - 1, where n is the size of the matrix.
2. For each i, add mat[i][i] (primary diagonal element) and mat[i][n-i-1] (secondary diagonal element) to the total sum.
3. After the loop, if n is odd, subtract mat[n//2][n//2] (middle element) from the total sum, as it’s been counted twice.
4. Return the total sum.

10 Prerequisite LeetCode Problems

Here are 10 problems that use similar underlying concepts:

1. Transpose Matrix: Given a matrix, return its transpose. This problem, like the original, involves iterating over a matrix and modifying elements based on their indices.

2. Diagonal Traverse: This problem involves traversing a matrix diagonally and requires an understanding of the relationship between row and column indices in a matrix, similar to the original problem.

3. Find Lucky Integer in an Array: You have to iterate over an array and make a decision based on the value and its frequency, much like you iterate over the matrix and make decisions based on the index in the original problem.

4. Toeplitz Matrix: This problem asks you to verify if a matrix is a Toeplitz matrix, which involves checking diagonal elements. It’s similar to our problem where we work with diagonal elements.

5. Reshape the Matrix: Reshaping a matrix involves understanding how elements in a matrix are indexed and how those indices change when the matrix is reshaped.

6. Spiral Matrix: This problem requires knowledge about how to iterate over a matrix in a spiral order. It’s related to our problem as it requires you to understand how to traverse a matrix in a non-traditional order.

7. Set Matrix Zeroes: This problem involves manipulating a matrix based on the values of its elements, similar to how we manipulated our matrix based on its indices.

8. Sort Array By Parity: In this problem, you need to sort the array such that all even integers come before all the odd integers. The pattern recognition in sorting the array is a strategy used in our original problem.

9. Matrix Cells in Distance Order: This problem requires understanding the distance between cells in a matrix, which can be linked to understanding the position of elements in the original problem.

10. Sort the Matrix Diagonally: The problem is about sorting the matrix diagonally which requires a deep understanding of how elements are positioned diagonally in a matrix - the same concept used in the original problem.