The parity (whether $$$n$$$ is odd or even) matters.

Swapping two adjacent cats keeps both of them close to their original location and changes both of their locations.

If $$$n$$$ is even, the optimal distance is $$$n$$$, and if $$$n$$$ is odd the optimal distance is $$$n+1$$$.

$$$i+j \leq 2 \cdot n$$$

The number of pairs $$$(a, b)$$$ such that $$$a \cdot b \leq x$$$ is $$$O(x log x)$$$.

If $$$n$$$ is even, the optimal distance is $$$n$$$, and if $$$n$$$ is odd the optimal distance is $$$n+1$$$.

What's the minimum value that an edge from $$$a$$$ to $$$b$$$ can be?

Use edges with negative value whenever you can.

The sum of the values of edges with positive weight must be $$$\geq$$$ the maximum value in the array.

The answer is always unique.

Try to figure out what the location of the $$$i$$$-th element would be if you only looked at the first $$$i$$$ elements, then the first $$$i+1$$$, etc. to find an $$$O(nq)$$$ solution.

Use sqrt decomposition to optimize it.