Permutation Cycle Queries

You are given an array $A$ of length $N$. For a permutation $P$ of the integers $1$ to $N$, let $f(P,A)$ denote the sum of the minimum values of $A_i$ within each cycle of the set of permutation cycles of $P$. A permutation cycle is a cycle in the permutation graph, where the nodes represent the indices $1$ to $N$, and the edges connect node $i$ to node $P_i$. If $P$ is a permutation, each connec