<p>There is a 2D <code>grid</code> of size <code>n x n</code> where each cell of this grid has a lamp that is initially <strong>turned off</strong>.</p>

<p>You are given a 2D array of lamp positions <code>lamps</code>, where <code>lamps[i] = [row<sub>i</sub>, col<sub>i</sub>]</code> indicates that the lamp at <code>grid[row<sub>i</sub>][col<sub>i</sub>]</code> is <strong>turned on</strong>. Even if the same lamp is listed more than once, it is turned on.</p>

<p>When a lamp is turned on, it <strong>illuminates its cell</strong> and <strong>all other cells</strong> in the same <strong>row, column, or diagonal</strong>.</p>

<p>You are also given another 2D array <code>queries</code>, where <code>queries[j] = [row<sub>j</sub>, col<sub>j</sub>]</code>. For the <code>j<sup>th</sup></code> query, determine whether <code>grid[row<sub>j</sub>][col<sub>j</sub>]</code> is illuminated or not. After answering the <code>j<sup>th</sup></code> query, <strong>turn off</strong> the lamp at <code>grid[row<sub>j</sub>][col<sub>j</sub>]</code> and its <strong>8 adjacent lamps</strong> if they exist. A lamp is adjacent if its cell shares either a side or corner with <code>grid[row<sub>j</sub>][col<sub>j</sub>]</code>.</p>

<p>Return <em>an array of integers </em><code>ans</code><em>,</em><em> where </em><code>ans[j]</code><em> should be </em><code>1</code><em> if the cell in the </em><code>j<sup>th</sup></code><em> query was illuminated, or </em><code>0</code><em> if the lamp was not.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/19/illu_1.jpg" style="width: 750px; height: 209px;" />
<pre>
<strong>Input:</strong> n = 5, lamps = [[0,0],[4,4]], queries = [[1,1],[1,0]]
<strong>Output:</strong> [1,0]
<strong>Explanation:</strong> We have the initial grid with all lamps turned off. In the above picture we see the grid after turning on the lamp at grid[0][0] then turning on the lamp at grid[4][4].
The 0<sup>th</sup>&nbsp;query asks if the lamp at grid[1][1] is illuminated or not (the blue square). It is illuminated, so set ans[0] = 1. Then, we turn off all lamps in the red square.
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/19/illu_step1.jpg" style="width: 500px; height: 218px;" />
The 1<sup>st</sup>&nbsp;query asks if the lamp at grid[1][0] is illuminated or not (the blue square). It is not illuminated, so set ans[1] = 0. Then, we turn off all lamps in the red rectangle.
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/19/illu_step2.jpg" style="width: 500px; height: 219px;" />
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 5, lamps = [[0,0],[4,4]], queries = [[1,1],[1,1]]
<strong>Output:</strong> [1,1]
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 5, lamps = [[0,0],[0,4]], queries = [[0,4],[0,1],[1,4]]
<strong>Output:</strong> [1,1,0]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>9</sup></code></li>
	<li><code>0 &lt;= lamps.length &lt;= 20000</code></li>
	<li><code>0 &lt;= queries.length &lt;= 20000</code></li>
	<li><code>lamps[i].length == 2</code></li>
	<li><code>0 &lt;= row<sub>i</sub>, col<sub>i</sub> &lt; n</code></li>
	<li><code>queries[j].length == 2</code></li>
	<li><code>0 &lt;= row<sub>j</sub>, col<sub>j</sub> &lt; n</code></li>
</ul>

class Solution:
    def gridIllumination(self, n: int, lamps: List[List[int]], queries: List[List[int]]) -> List[int]:
        # n x n 2D grid.
        # Each cell, lamp is initially turned off 
        # When a lamp is turned on, it illuminates its cell and all other cells in the same row/col/diagonal.
        # Given 2D array queries, where queries[j] = [row_j, col_j] and determen whether grid[row_j, col_j] is illuminated or not.

        # Initialize dictionaries to keep track of illuminated rows, columns, and diagonals
        rows = defaultdict(int)
        cols = defaultdict(int)
        diag1 = defaultdict(int)  # r + c
        diag2 = defaultdict(int)  # r - c
        lamps_set = set()

        # Turn on lamps
        for r, c in lamps:
            if (r, c) not in lamps_set:
                rows[r] += 1
                cols[c] += 1
                diag1[r + c] += 1
                diag2[r - c] += 1
                lamps_set.add((r, c))

        def is_illuminated(r, c):
            return rows[r] > 0 or cols[c] > 0 or diag1[r + c] > 0 or diag2[r - c] > 0

        def turn_off_adjacent(r, c):
            for dr in [-1, 0, 1]:
                for dc in [-1, 0, 1]:
                    nr, nc = r + dr, c + dc
                    if 0 <= nr < n and 0 <= nc < n and (nr, nc) in lamps_set:
                        lamps_set.remove((nr, nc))
                        rows[nr] -= 1
                        cols[nc] -= 1
                        diag1[nr + nc] -= 1
                        diag2[nr - nc] -= 1

        # Process queries
        ans = []
        for r, c in queries:
            ans.append(1 if is_illuminated(r, c) else 0)
            turn_off_adjacent(r, c)

        return ans
        # Return ans, where ans[j] = 1 if query[j] is illuminated or not.
        
        # Handling diagonals.
        # n x n diagonal space can be converted into a (2n - 1) x (2n - 1) diagonal space.
        def grid_to_diagonal(row, col, n):
            """
            Transform grid coordinates to diagonal coordinates.
            
            :param row: Row in the original grid
            :param col: Column in the original grid
            :param n: Size of the original grid (assuming it's square)
            :return: Tuple of (diagonal_row, diagonal_col)
            """
            diag_row = row + col
            diag_col = n - 1 - row + col
            return diag_row, diag_col

        # Build Illuminated rows, cols, diag_rows, diag_cols
        illuminated_rows = defaultdict(int)
        illuminated_cols = defaultdict(int)
        illuminated_diag_rows = defaultdict(int)
        illuminated_diag_cols = defaultdict(int)
        lamp_set = set()

        for (row, col) in lamps:
            diag_row, diag_col = grid_to_diagonal(row, col, n)
            illuminated_rows[row] += 1
            illuminated_cols[col] += 1
            illuminated_diag_rows[diag_row] += 1
            illuminated_diag_cols[diag_col] += 1
            lamp_set.add((row, col))

        directions = [
            (-1, -1), (-1, 0), (-1, 1),
             (0, -1),  (0, 0),  (0, 1),
             (1, -1),  (1, 0),  (1, 1),
        ]

        # Query them & update queries in-place.
        answer = []
        for idx, (row, col) in enumerate(queries):
            diag_row, diag_col = grid_to_diagonal(row, col, n)

            if (illuminated_rows[row] > 0 or illuminated_cols[col] > 0 or 
                illuminated_diag_rows[diag_row] > 0 or illuminated_diag_cols[diag_col] > 0):
                  # queries[idx] = 1
                answer.append(1)
            else:
                # queries[idx] = 0
                answer.append(0)
        

            # After answering query, turn of 8 adjacent lamps and lamp at query.
            for dy, dx in directions:
                y, x = row + dy, col + dx
                if 0 <= y < n and 0 <= x < n and (y, x) in lamp_set:
                    y_diag, x_diag = grid_to_diagonal(y, x, n)
                    illuminated_rows[y] -= 1
                    illuminated_cols[x] -= 1
                    illuminated_diag_rows[y_diag] -= 1
                    illuminated_diag_cols[x_diag] -= 1
                    lamp_set.remove((y, x))
        
        return answer
  

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