Help me understand your day 19 solution

[AndrewSav]

Hey, I'm looking at https://github.com/erik-adelbert/aoc/blob/main/2021/19/aoc19.go and I'm wondering how it works. In align, you sort by each of three axes separately and then you take differences for adjacent points. Since the known set only grows, it is not apparent why any two points that appears in two scanners result sets should be adjacent on any of the three axes. It is quite likely when known result set is small but becomes less and less likely as it grows. I can see that the program achieves its goal, but I would like to understand if this is a coincidence, or is it always guaranteed to succeed, If this is the latter, then why?

[erik-adelbert]

Hi @AndrewSav, sorry for the delay we are looking for points (beacons) that are the same regardless of their coordinate system (scanners). To do so, we're using the fact that any vector field of a point set is the same regardless of the coordinate system origin: while orientation is considered fixed (by rotation), we're taking the shortcut of only verifying successive differences along each axis for any match. If we find one, we use it as a temporary origin and rebase the candidate set on it.
By now, you should see that it amounts to superpose the candidate set with the fixed one and that there will easily be 12 matches if the alignment is correct (by design of the challenge and its input).
To answer your question, sorting along axis is just a trick to predictably build the same-ish vector field.
And yes, a real question may be how to devise the correct threshold (here 12)?