Make Grid Comparable

You are given a grid with $2$ rows and $N$ columns containing **distinct** integers. The number in the $i$-th row and $j$-th column is denoted by $A_{i, j}$. In one operation, you can swap numbers in any two cells which share a side. For a binary string $S$ of length $N$, let's define $f(S)$ as follows: * $f(S)$ is the smallest number of swaps required to achieve a state of board in which $A_{1,