Find a Peak Element II

Find a Peak Element II

What is this problem about?

The Find a Peak Element II coding problem is a multidimensional extension of the classic peak finding challenge. Given a 2D matrix where no two adjacent cells are equal, a "peak" is an element that is strictly greater than all its immediate neighbors (up, down, left, and right). Your task is to find the coordinates of any one such peak element in $O(N \log M)$ or $O(M \log N)$ time.

Why is this asked in interviews?

This problem is a favorite at Microsoft and Google because it tests your ability to optimize a search in a 2D space. A brute-force scan would take $O(N imes M)$, but the logarithmic requirement forces you to apply the Binary Search interview pattern in a clever way. It evaluations whether you can reduce a complex 2D search into a 1D decision process by identifying properties that guarantee the existence of a solution in a specific sub-region.

Algorithmic pattern used

This problem uses a combination of Binary Search and Global Maximum finding.

1. Pick a middle column.
2. Find the maximum element in that specific column (which takes $O(N)$ time).
3. Compare this global column maximum with its neighbors in the adjacent columns (left and right).
4. If it's greater than both, it's a peak.
5. If the neighbor to the left is larger, a peak must exist in the left half of the matrix.
6. If the neighbor to the right is larger, a peak must exist in the right half.
7. Repeat the process on the chosen half.

Example explanation

Imagine a $3 imes 3$ matrix:

[ 1, 4, 3 ]
[ 2, 5, 2 ]
[ 3, 6, 4 ]

1. Choose middle column (Index 1): [4, 5, 6].
2. Max in this column is 6 (at row 2, col 1).
3. Compare 6 with neighbors: 3 (left) and 4 (right).
4. Since 6 > 3 and 6 > 4, 6 is a peak element.
5. Return [2, 1].

Common mistakes candidates make

• Applying 1D binary search incorrectly: Trying to binary search every row independently, which doesn't guarantee a peak relative to vertical neighbors.
• Complexity errors: Implementing an $O(N imes M)$ solution when $O(N \log M)$ is required.
• Boundary conditions: Not correctly handling cases where the "middle" column is at the very edge of the matrix.

Interview preparation tip

Whenever you need to solve a 2D problem in better than linear time, think about "Dimensional Reduction." Can you solve a 1D version of the problem at a specific index and use that result to eliminate half of the remaining 2D space? This is a core Matrix interview pattern.