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

As Peter says, it would be ideal to minimise the size of each integer — but in fact, you can do it without putting restrictions on the number of items. Variable-byte encoding is a way of compressing sequences of integers by only using the necessary number of bytes. The most common way of doing this is to reserve one bit in each byte to indicate whether that byte is the last one in the current list item.

It could be useful to use delta encoding first. That's where you store the differences between the integers, rather than the integers themselves — meaning that they end up compressing better with variable-byte. Of course, the integers being stored (perhaps the IDs of items being changed, in your case) would have to be sorted first, but that doesn't seem like it'd be a problem for you.