Trench in 2-d array

Question

Suppose there is an nxn array. How to find a pair of indices i and j, such that;

A[i][j] < A[i+1][j], A[i][j] < A[i-1][j], A[i][j] < A[i][j+1],A[i][j] < A[i][j-1]

All I was able to think was an O(n ² ) algo, which loops across the complete array, and finds the element based on the given conditions.

Can we have a better solution?

Answer 1

Since there is no ordering to the matirx, It is immpossible to confirm existence or non-existence of the index without explicitly checking every index at least once.

By that logic, I am pretty sure this problem should have a lower bound of O(n^2)