Minimize the Sum

You are given an $N \times N$ grid. Each cell in the grid contains a tree with a positive integer height. You also have a machine that operates with a positive integer parameter $K$. When you apply the machine to a row, every tree in that row with height greater than $K$ will have its height reduced to exactly $K$; trees with height less than or equal to $K$ remain unchanged. For any row $i$ (w