Problem Description

The task is to compute a matrix answer based on another 2D grid given as input. The grid is a two-dimensional array with m rows and n columns, and we want to create a new 2D array answer of the same dimensions.

For each cell in the answer matrix at position (r, c), the value is determined as follows:

• topLeft[r][c]: count the number of distinct values that appear in the cells on the top-left diagonal of cell (r, c) in the original grid. This diagonal includes cells (x, y) from the top row or the leftmost column cutting through to just above (r, c) such that x < r and y < c.

• bottomRight[r][c]: count the number of distinct values that appear in the cells on the bottom-right diagonal of cell (r, c) in the original grid. This diagonal includes cells (x, y) starting just below (r, c) running down to the bottom right corner of the grid such that x > r and y > c.

The value in answer[r][c] is the absolute difference between these two counts: |topLeft[r][c] - bottomRight[r][c]|.

The problem requires returning the matrix answer, which is constructed by performing these computations for every cell (r, c) of grid.

Intuition

The intuition behind the solution is to focus on each cell (r, c) individually and to accumulate the distinct values along its top-left and bottom-right diagonals. Using sets is helpful because sets naturally maintain unique elements, thereby making the counting of distinct values straightforward.

For the top-left diagonal, we start from the current cell (r, c) and move upwards and to the left (x - 1, y - 1) until we either reach the first row or the first column.

For the bottom-right diagonal, we start from the current cell (r, c) and move downwards and to the right (x + 1, y + 1) until we either reach the last row or the last column.

At each step for both diagonals, we add the values of grid[x][y] to their respective sets. Once all unique values are accumulated, the sizes of the sets represent the count of distinct values in each diagonal. We calculate the absolute difference of these counts and assign it to answer[r][c].

This process is repeated for every cell in the grid, which eventually yields the completed answer matrix. The brute force nature of this solution is acceptable given the problem constraints, and it directly translates the problem's requirements into algorithmic steps to find the solution.

Solution Approach

The solution as implemented above employs a straightforward, brute-force approach. To compute the answer[r][c], the algorithm inspects two distinct diagonals for each cell (r, c) in the grid. Here is a step-by-step breakdown of the approach:

1. Initialize a matrix ans with the same dimensions as grid but filled with zeros. This will store the final results.

2. Traverse every cell (r, c) in the grid. Use nested loops, the outer loop running through the rows, and the inner loop going through the columns.

3. To find topLeft[r][c], initialize an empty set s. Then, iterate from the current cell upwards and leftward diagonally (decreasing both row and column indices). For each cell encountered, add the value of the cell from grid[x][y] to the set s. The loop stops when reaching the top row or the leftmost column. After iteration, tl is set as the length of the set s, which is the count of unique values in the top-left diagonal.

   x, y = i, j
   s = set()
   while x and y:
       x, y = x - 1, y - 1
       s.add(grid[x][y])
   tl = len(s)

4. Repeat a similar process to find bottomRight[r][c]. Initialize another empty set s, and iterate from the current cell downwards and rightward diagonally (increasing both row and column indices). Add the value from grid[x][y] to set s until reaching the bottom row or the rightmost column. Assign br as the length of this set.

   x, y = i, j
   s = set()
   while x + 1 < m and y + 1 < n:
       x, y = x + 1, y + 1
       s.add(grid[x][y])
   br = len(s)

5. Calculate the absolute difference between tl and br and assign it to ans[i][j]. This step executes for every cell (r, c), filling the ans matrix.

   ans[i][j] = abs(tl - br)

6. After filling in all values, return the ans matrix.

Using this method, every cell in the grid is visited, and its corresponding diagonals are processed to determine the values of answer[r][c]. The utilization of sets for counting unique elements is pivotal because it eliminates the need for manual checks for duplicates, simplifying the code and ensuring accuracy.

Example Walkthrough

Let's go through a small example to illustrate the solution approach.

Suppose we have the following grid matrix:

grid = [
    [1, 2, 3],
    [4, 1, 6],
    [7, 8, 1]
]

We want to construct an answer matrix where each element is the absolute difference between the count of distinct numbers in its top-left and bottom-right diagonals.

Step-by-step Process:

1. Initialize an answer matrix with zeros having the same dimensions as grid:

answer = [
    [0, 0, 0],
    [0, 0, 0],
    [0, 0, 0]
]

2. Start iterating over each cell (r, c) of the grid. First with cell (0, 0):
   • There is no top-left diagonal, so topLeft = 0.
   • For the bottom-right diagonal, only 1 is on the diagonal, so bottomRight = 1.
   • answer[0][0] = |0 - 1| = 1
   • Update the answer matrix:

answer = [
    [1, 0, 0],
    [0, 0, 0],
    [0, 0, 0]
]

3. Move to cell (0, 1):
   • Again, no top-left diagonal, so topLeft = 0.
   • The bottom-right diagonal has 1 and 6, so bottomRight = 2.
   • answer[0][1] = |0 - 2| = 2
   • Update the answer matrix:

answer = [
    [1, 2, 0],
    [0, 0, 0],
    [0, 0, 0]
]

Now we'll do the same for cell (1,1), which has both top-left and bottom-right diagonals:

4. Inspect the cell (1, 1):
   • The top-left diagonal contains 4 and 1 (so topLeft[r][c] = 2).
   • The bottom-right diagonal contains only 1, hence bottomRight[r][c] = 1.
   • answer[1][1] = |2 - 1| = 1.

Update the answer matrix:

answer = [
    [1, 2, 0],
    [0, 1, 0],
    [0, 0, 0]
]

5. Continue the process for each cell, calculating the unique counts on both diagonals and computing their absolute difference.

After completing this process for all cells, we would obtain the final answer matrix:

answer = [
    [1, 2, 1],
    [1, 1, 0],
    [1, 0, 0]
]

Each step of the process follows the explanation in the solution approach, utilizing the property of sets to count unique elements along the top-left and bottom-right diagonals for each cell and calculating the absolute difference between these unique counts for the answer matrix.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code provided is primarily determined by the two nested loops, each iterating over the dimensions of the grid, and the two while loops inside the nested loops. The outer loops run m * n times where m is the number of rows and n is the number of columns in the grid (m and n can be considered separately if not square). The inner while loops iterate at worst from 0 to i for the top-left diagonal elements, and from i to m for the bottom-right diagonal elements (and similarly for j and n). This gives us two arithmetic progressions counting down and up respectively. The sum of operations due to these internal while loops corresponds to the sum of two arithmetic series, which is effectively O(m + n) for each outer loop iteration. Therefore, the total time complexity is roughly O((m * n) * (m + n)), which can also be represented as O(m^2 * n + m * n^2) when m and n are different.

Space Complexity

The space complexity is O(m * n) due to the auxiliary space required for the ans list which is the same size as the grid. Additionally, the set s takes up space, but since it is overwritten in each iteration of the loops rather than accumulated, the maximum space it uses at any given time is the size of the largest number of distinct elements on a diagonal, which is at most min(m, n). This does not change the overall space complexity, which remains O(m * n).

Learn more about how to find time and space complexity quickly using problem constraints.