How to compute the absolute minimum amount of changes to convert one sortorder into another?

Question

Goal

How to encode the data that describes how to re-order a static list from a one order to another order using the minimum amount of data possible?

Answer 1

A quick fix might be to use a Zobrist hash to spot cases where you go back to a prior order. That is, after each swap, calculate a hash based on the permutation you reach. Each hash maps to the shortest sequence of swaps found so far for that particular permutation.

This can easily be extended with a bit of exploratory searching - the Zobrist hash was invented as a way to optimise game tree searches.

It's easy to give a strict lower bound to the number of swaps, of course - the number of items that are not in their required locations. Whether that lower bound is actually achievable, though, is a more difficult problem.