I tend to remember techniques more than specific algorithms, but:
One really fun algorithm involved optimizing an n^2 naive tree algorithm to O(n), ignoring logs.
For me, the way I reasoned about it was expanding the number of objects to consider (n^3), then there were observations you could apply to bring it down to O(n).
If you asked me what exactly the correspondence was between the final reduction and the original problem statement, I couldn't tell you. Maybe there was a more direct way to get to the solution.
But that style of thinking carries on with me to real tasks too. Sometimes it's easier to simplify, other times it might actually be easier to complexify, as long as you trust that you're not going down "arbitrary" complexity.
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