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Yes. Arphic, for instance, calls them radical-based fonts.

Fully algorithmically? I have no idea, as I'm not really in the fonts business.

But I'm pretty sure they're not actually redrawing every character from scratch, and are actually reusing the subcomponents (at very least for normal fonts). But how much of that is actually automated - you'd have to ask actual font designers.

Droid Sans Fallback used this strategy during Android 1.x-2.x era. Its successor Noto Sans don't do the same.