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Only the ennemies? And the yarns?
- Enemy positions, Yarn, Relics, and destructible boxes shuffled, bosses are not switched though. Dwellers and Spiders are exceptions as most of the time they are required where they are.
- Time piece levels are randomized; either in a fair order where each hat is possible to get before needed, or just starting with all hats and the Hookshot Badge. I think Finales would have to still be set in stone as I'm not sure how the Battle of the Birds levels would even work.
- Time piece appearance location randomized could be interesting. (Exception being bosses because they'd be impossible to get.)
If you really want to go all out you could even shuffle the objects between different levels. (Security guards and cameras could be found in Subcon Forest, Hats could be retrieved in different levels than normal, etc.)
Randomizing AHiT would be much more involved than Mario randomizers though, simply because the player doesn't have all movement options available at the start. It would have to be handled like a Zelda randomizer where required collectibles (yarn, hookshot) are "ordered" in the random level unlocks to make it possible to finish.
Such a program would need a model of how Hat Kid's movement abilities work, in particular how far she can move with each use of them and what limits are needed on the space to be able to use them (eg, if she can jump 3m horizontally at a given height, but needs 2m of vertical space to do it in, then that dictates a minimum ceiling height of 2m).
Normal jump, double jump, wall run, dive, back-and-forth wall run and jump, grappling hook. I think that's it. With a model of how each ability affects movement and a library of rift components, it could generate a series of movements and the appropriate components needed to create a path through 3D space.
The main obstacle to making such a program would be needing to know how Unreal Engine 3 structures a level file and what specific things A Hat In Time needs in its levels. If you had good documentation on those, you could probably cobble together a command-line program to take a handful of parameters (allowed movement types, how horizontal and vertical to make it, whether to include switch puzzles, how long to make the level, minimum number of paths, and of course a random seed) and spit out a level file without too much trouble.