Given a permutation $P$ of an unknown array $U$ of length $N$ and a function $f(Q)$ that calculates the minumum number of swaps between consecutive elements of array $Q$ to reach $U$, what is the minimum number of invocations of $f$ to discover $U$, worst case?
An upper bound of $N$ invocations is at hand:
Calculate $F(P)$, then for each $i,2\le i\le N$ move $P_i$ to the start of array $P$, then calculate $f(P)$ to discover the correct position of $P_i$ in subarray $P_{1...i}$. Being a little more careful, we can spare the last calculation of $f(P)$ since having $P_{1...N-1}$ ordered and knowing $f(P)$ is enough to calculate the correct position of $P_N$.
This yields the upper bound of $N-1$ invocations
For an array with length $2$, is clear that we need at least one invocation so this is a tight bound.
Question: For a sufficiently large $N$, is it possible to do less than $N-1$ invocations of $f$ in the worst case?.
Asymptotically, since the possible outputs of $f$ are $0...N(N-1)/2$ by the pidgeonhole principle we need at least $O(\log_{N(N-1)/2}(N!))=O(\log_N(N!))=O(N)$ (by Stirling's approximation) invocations to discover the array $P$. So this prompts a second question:
Question: For a sufficiently large $N$, is it possible to do less than $c\times N$ invocations of $f$ for some constant $c<1$ in the worst case?.
