Problem Description

You are tasked with finding the largest '+' shape that can be formed in a binary grid, where '1's represent a potential part of the '+' and '0's represent a space where part of a '+' cannot exist. The grid is n x n in size, and originally it's filled with '1's. However, there are some coordinates provided in the mines array which represent where '0s' will be, thus creating obstacles for forming the '+'. The order of the plus sign is defined as the length from its center to its edge, which means the total size will be related to the order as order*2 - 1. The goal is to return the highest order of a '+' sign found in this grid. If no '+' can be formed, the function should return 0.

Intuition

When tasked with finding the largest axis-aligned plus sign of '1's in a modified binary grid, we need to think about what defines a plus sign. For a plus sign to exist at any location, there must be uninterrupted sequences of '1's extending to the left, right, up, and down from that center '1'.

The intuition behind finding the largest plus sign involves assessing each potential center for how far in each of the four directions (up, down, left, and right) we can go before hitting a '0' or the edge of the grid.

We use dynamic programming to avoid redundant calculations. A matrix dp of the same dimensions as grid is created. Each element in dp will eventually represent the largest order of a plus sign that can have that cell as its center.

We initialize dp with 'n', representing the maximum possible order for a plus sign in an n x n grid. We then iterate through the indices in the mines array, setting the corresponding dp cell to '0', indicating an obstacle.

Next, we make four passes through the grid for each cell (i, j):

• From left to right: if there's no obstacle, we increase the count left, otherwise we start the count over. We then set dp[i][j] to the minimum of its current value (which will be at least left).
• From right to left, following the same logic with the variable right.
• From top to bottom and bottom to top: the same process follows for the variables up, and down.

After this process, we know for any cell in dp, it holds the minimum possible extension in all four directions, effectively giving us the order of the largest plus sign centered at that cell.

Finally, we look for the maximum value in the dp array. This value is the order of the largest axis-aligned plus sign of '1's in the grid.

Learn more about Dynamic Programming patterns.

Solution Approach

To implement the solution for finding the largest plus sign, we utilize the dynamic programming technique that involves creating a 2D array called dp that serves as a table to store intermediate results. This is crucial because it will help in reducing the complexity by storing the achievable order of a plus sign at each (i, j) position, thus avoiding recomputation.

Here's the step-by-step approach:

1. Initialize the dp matrix, with n rows and n columns, setting every cell to n. This pre-population assumes the plus sign can extend to the edge of the grid in every direction from any cell.

2. Iterate over the mines list, which contains the positions that must be set to '0'. The mines list modifies the dp matrix by setting dp[x][y] = 0 for every [x, y] pair, showing that no plus sign can be centered at these 'mined' cells.

3. To calculate the order of the largest plus sign that can be centered at each cell, iterate over each row and column four times, once for each direction: left, right, up, and down.

   • For left and rightward checks, we loop through each row, once from left to right and then in the reverse direction. If dp[i][j] is not zero, we increment left or right as appropriate, showing that the arm of the plus can extend in that direction.

   • Similarly, for upwards and downwards, we loop through each column in both top-to-bottom and bottom-to-top directions, incrementing up or down if dp[j][i] is not zero.

   • At each step, we update the dp matrix by assigning the minimum value amongst the current dp value and the 'left', 'right', 'up', or 'down' counters.

This process ensures that dp[i][j] holds the order of the largest plus sign for which the cell (i, j) can be the center, considering the constraints in all four directions.

4. After the conclusion of the loops, scan through the dp matrix to find the maximum value. This value is the order of the largest plus sign that can be found in the grid.

The specific data structure used here is a 2D list (list of lists) for the dp table, enabling direct access to each cell for its up, down, left, right sequence calculations.

In terms of the algorithm, the solution employs an efficient linear scan of rows and columns, taking O(n^2) time — a significant optimization over brute force approaches that would require O(n^3) or greater. It effectively combines the principles of dynamic programming with those of matrix traversal, effectively reducing the problem to simple arithmetic and comparisons.

Example Walkthrough

Let's say we have a 3x3 grid (n = 3) and the mines array is [[1, 1]], meaning there will be a '0' in the center of the grid.

Here's a walkthrough of applying the solution step-by-step to this grid:

1. We initialize the dp matrix to represent the maximum possible order of the plus sign in a 3x3 grid:

dp = [[3, 3, 3],
      [3, 3, 3],
      [3, 3, 3]]

2. We then apply the mines: since we have a mine at [1, 1], we set dp[1][1] to 0, indicating no plus can be centered here.

dp = [[3, 3, 3],
      [3, 0, 3],
      [3, 3, 3]]

3. Next, we make passes to calculate the left, right, up, and down order for each cell:

   • Left to right pass: We count the number of consecutive '1's from the left for each cell in a row and update the dp value.

     After the left to right pass:

     dp = [[1, 2, 3],
           [1, 0, 3],
           [1, 2, 3]]

   • Right to left pass: Similar to above, but from the right.

     After the right to left pass:

     dp = [[1, 2, 3],
           [1, 0, 1],
           [1, 2, 3]]

     (We only update if the value of the current cell is greater than the count.)

   • Top to bottom pass: We do the same vertically.

     After top to bottom pass:

     dp = [[1, 1, 1],
           [1, 0, 1],
           [1, 2, 1]]

   • Bottom to top pass: And finally, from bottom to top.

     After bottom to top pass:

     dp = [[1, 1, 1],
           [1, 0, 1],
           [1, 1, 1]]

4. Now dp contains the order of the largest '+' that can be formed with each cell as its center. We scan through dp to find the maximum value which will give us the order of the largest '+'. In our example:

   • The maximum value in dp is 1, which means the largest possible plus sign order is 1, and the size of this plus sign would be 1*2 - 1 = 1. Hence, the largest plus sign we can form has arms of length 1.

To summarize, for this 3x3 grid with a mine at the center, the largest '+' sign we can form has an order of 1, extending a single cell in each direction from its center.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be analyzed by breaking down each part of the code:

1. Initializing the dp matrix with the size n x n takes O(n^2) time.
2. The first loop to set the mines takes O(m) time, where m is the number of mines.
3. The nested loops to calculate the left, right, up, and down values take O(n^2) time. For each of the n rows and n columns, the loop runs n times.
4. The final nested loops also run in O(n^2) time, since it involves traversing the dp matrix again to calculate the minimum for each position and also to find the maximum order for plus sign at the end.

Overall, the time complexity is dominated by the nested loops, so the final time complexity is O(n^2).

Space Complexity

The space complexity is determined by the extra space used by the algorithm, which is:

1. The dp matrix that requires O(n^2) space.
2. Four variables left, right, up, and down are used but they only add O(1) space.

Thus, the space complexity of the algorithm is O(n^2).

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