Maximum Score From Grid Operations

The problem requires calculating the maximum possible score by applying selective column operations in a 2D grid. Each operation changes white cells to black in a column up to a specific row, affecting the scoring pattern. Using state transition dynamic programming allows systematically exploring all sequences while avoiding redundant calculations and ensuring the optimal sum is obtained.

Problem Statement

You are given an n x n 2D grid of integers where all cells start as white. You can perform operations to color cells black in a column from the top down to a chosen row. Each operation affects the scoring potential of adjacent white cells.

The score is computed as the sum of grid[i][j] for each white cell that has a horizontally adjacent black cell. Determine the maximum score achievable after applying any number of column coloring operations, optimizing the selection of rows for each column.

Examples

Example 1

Input: grid = [[0,0,0,0,0],[0,0,3,0,0],[0,1,0,0,0],[5,0,0,3,0],[0,0,0,0,2]]

Output: 11

In the first operation, we color all cells in column 1 down to row 3, and in the second operation, we color all cells in column 4 down to the last row. The score of the resulting grid is grid[3][0] + grid[1][2] + grid[3][3] which is equal to 11.

Example 2

Input: grid = [[10,9,0,0,15],[7,1,0,8,0],[5,20,0,11,0],[0,0,0,1,2],[8,12,1,10,3]]

Output: 94

We perform operations on 1, 2, and 3 down to rows 1, 4, and 0, respectively. The score of the resulting grid is grid[0][0] + grid[1][0] + grid[2][1] + grid[4][1] + grid[1][3] + grid[2][3] + grid[3][3] + grid[4][3] + grid[0][4] which is equal to 94.

Constraints

• 1 <= n == grid.length <= 100
• n == grid[i].length
• 0 <= grid[i][j] <= 109

Solution Approach

Define the DP State

Use a dynamic programming array where dp[i][mask] represents the maximum score achievable for the first i columns with the current column state represented by mask. The mask encodes which rows are blackened, allowing efficient state transitions.

State Transitions

Iterate over all columns and possible masks. For each mask, simulate applying an operation on any row in the current column, update the mask for the next column, and compute the score contribution from newly white cells adjacent to black cells.

Compute Maximum Score

After processing all columns, the maximum value across all final masks in dp[n] gives the result. This approach avoids recomputation by caching intermediate states and ensures the optimal sequence of operations is found.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the number of columns n and the possible masks for rows, roughly O(n * 2^n), and space complexity is proportional to the number of masks stored for DP, also O(2^n * n).

What Interviewers Usually Probe

• Expect an efficient state representation for column operations.
• Look for correct handling of row masks and adjacent white cell scoring.
• Check that DP avoids recomputing identical column sequences.

Common Pitfalls or Variants

Common pitfalls

• Ignoring the effect of previous operations on scoring white cells.
• Not properly encoding or updating masks, leading to incorrect DP states.
• Overlooking edge cases where no operations are optimal for certain columns.

Follow-up variants

• Compute maximum score when operations can also color entire rows.
• Allow diagonal adjacency instead of only horizontal for scoring.
• Consider grids where some cells cannot be blackened due to constraints.

FAQ

What is the core dynamic programming pattern used in Maximum Score From Grid Operations?

It uses state transition DP where each state represents a column and the blackened rows, tracking score impact efficiently.

How do I determine which row to apply an operation on in a column?

Consider all possible rows for each column and use DP to select the row that maximizes the total score based on previous column states.

Can this problem be solved greedily instead of DP?

No, because each operation affects future scoring, a greedy approach may miss optimal sequences. State transition DP ensures correctness.

What is the expected time complexity for this problem?

Time complexity is roughly O(n * 2^n) due to iterating over all column masks and rows for state transitions.

Are there any constraints on grid values I should be aware of?

Yes, each grid[i][j] is between 0 and 10^9, and n ranges from 1 to 100, which affects mask size and DP feasibility.