seek distance then amounts to finding that permutation, $P$, of 
\{1,2,...,$N$\} that minimizes
\begin{equation}
E[P] =  \sum_{i=1}^N \sum_{j=1}^N |P(i) - P(j)| \pi_i \rho_{ij}, \label{eqn:goal1}
\end{equation}
and then remapping the cylinders according to that permutation.  Note
that any such permutation
remapping can be accomplished by a sequence of simple pairwise exchanges of cylinder
contents.

Nevertheless, the number of permutations to be considered in
minimizing (\ref{eqn:goal1}) is $N$!,
where a reasonable value of $N$ is 2000.  Since 
2000! $\approx 10^{5736}$, which dwarfs the number of
atoms in the universe, the problem of finding
this optimum permutation is clearly intractable.
