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

Another possible solution, ignoring your data structure...

Send a set of IDs/indexes for items that have changed (if it's a completely random sparse subset, just list them) and a permutation number describing the re-ordering of that subset. The permutation number will need a big integer representation - size should be proportional to log(n!) where n is the number of items changed.

The permutation number is defined from a permutation array, of course, but this detail can be avoided when decoding. The trick is to encode the permutation number so that, once you have swapped the correct first item into the first slot, you can also derive a new permutation number which is correct for the tail of the array.

That is...

while not empty(indexes)
  item-to-swap := permutation-no remainder len(indexes)
  permutation-no := permutation-no div len(indexes)
  if item-to-swap != 0 : swap slot[indexes[0]], slot[indexes[item-to-swap]]
  indexes := tail(indexes)

The != 0 check is needed even though all items needed changing at the start - an item might have been swapped upwards into it's correct location earlier in the loop.

This doesn't attempt to optimise the number of swaps - an item may be swapped upwards several times before being swapped downwards into it's correct location. That said, the permutation number is probably an optimum-space representation for a random permutation of an array. Given that your permutation only affects a small subset of the full array, using the smaller permutation number for that subset makes a lot of sense.