Largest 1-Bordered Square

Largest 1-Bordered Square

1. What is this problem about?

The Largest 1-Bordered Square interview question involves finding the largest square subgrid within a binary matrix (containing only 0s and 1s) such that all the elements on its borders are 1s. The interior of the square can contain any combination of 0s and 1s; only the top, bottom, left, and right edges must consist entirely of 1s. The goal is to return the total number of elements in such a square (i.e., its area, $side^2$) or simply the side length.

2. Why is this asked in interviews?

Companies like Samsung and Amazon ask this Largest 1-Bordered Square coding problem to test a candidate's ability to optimize a grid-based search. A brute-force approach would involve checking every possible square at every possible position, which is highly inefficient. The problem rewards candidates who can use preprocessing techniques—specifically Dynamic Programming—to transform a complex geometric check into a constant-time operation. It demonstrates spatial reasoning and efficiency in handling 2D data structures.

3. Algorithmic pattern used

The optimal Array, Matrix, Dynamic Programming interview pattern uses two auxiliary 2D arrays (or one 2D array of pairs) to store the count of consecutive 1s ending at each cell. One array top[r][c] stores the number of consecutive 1s above (and including) the current cell, and left[r][c] stores the number of consecutive 1s to the left. Once these are computed, you iterate through each cell (treating it as the bottom-right corner of a potential square) and check possible side lengths $d$. A square of side $d$ exists if the current cell has at least $d$ 1s to the left and $d$ 1s above, AND the other two corners (top-right and bottom-left) also satisfy the boundary conditions.

4. Example explanation

Imagine a $3 \times 3$ matrix:

1 1 1
1 0 1
1 1 1

The largest 1-bordered square here has side 3.

• Bottom-right corner is (2,2). left[2][2] is 3, top[2][2] is 3.
• To confirm side 3, we check the top-right corner (0,2) for its left count: left[0][2] is 3.
• We check the bottom-left corner (2,0) for its top count: top[2][0] is 3. Since all conditions are met, the side length is 3.

5. Common mistakes candidates make

• Checking all cells: Trying to verify the borders by iterating through every element of the border for every candidate square. This leads to $O(N^3 \times N)$ or $O(N^4)$ complexity.
• Wrong DP transition: Miscalculating the prefix counts of 1s.
• Incomplete border check: Only checking the bottom and right edges of a square but forgetting to verify the top and left edges.
• Integer overflow: While not common for side lengths, failing to handle empty matrices or 1x1 cases can lead to errors.

6. Interview preparation tip

Whenever you face a matrix problem involving "squares" or "rectangles" with specific properties, think about how preprocessing can help. Precomputing "how many 1s are to the left" or "how many 1s are above" is a standard trick that reduces the complexity of verifying a shape's properties significantly.