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If tried that in eg Haskell to see if I can beat the standard library's highly optimized sort.

In theory, you can get O(n log k) performance, where k is the number of distinct elements (so k <= n). And crucially: you don't need to know k up-front.

In practice, all my attempts were absolutely slower than the standard approach based on binary comparisons only. (But that's saying more about me than about the domain.)

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