Find local minimum in n x n matrix in O(n) time

Question

So, this is not my home work question, but it is taken from an ungraded homework of the coursera course on algorithms and data structures (which is now complete).

Y

Answer 1

Well, this is how you divide and conquer it.

1) Divide the n x n matrix into four n/2 x n/2 sub matrices.

2) Keep dividing the sub matrices recursively until you end up with a 2 x 2 matrix

3) Check if any element of 2 x 2 matrix is a local Minimum.

Recurrence equation is : T(n) = 4*T(n/2) + O(1)

4*T(n/2) for 4 n/2 x n/2 sub matrices and O(1) for checking if 2 x 2 sub matrix has a local minimum

Master theorem says this is a O(n^2) worst case bound.

But I think we can get a best case O(n) bound,

(" RED ALERT! ---BEST CASE IS BOGUS, JUST BOGUS---RED ALERT!").

if we exit the recursion stack after we have have found a local minimum in step 3.

Pseudocode:

private void FindlocalMin(matrix,rowIndex,colIndex,width){
    if(width == 1){ checkForLocalMinimum(matrix,rowIndex,colIndex); return;} //2x2 matrix
    FindlocalMin(matrix,rowIndex,colIndex,width/2);  
    FindlocalMin(matrix, (rowIndex + (width/2) + 1) ,colIndex,width/2);
    FindlocalMin(matrix,rowIndex, (colIndex + (width/2) + 1) ,width/2);
    FindlocalMin(matrix,(rowIndex + (width/2) + 1), (colIndex + (width/2) + 1) ,width/2);
}

private void checkForLocalMinimum(.........){
    if(found Local Minimum in 2x2 matrix){ exit recursion stack;} 
}

Here is a java implementation