Computing 3SUM problem in $O(n\lg n) + \frac{n^2}{4}$ time
There are known algorithms that take strictly less than $O(n^2)$ time (i.e., they run in $o(n^2)$ time), under certain theoretical computational models.  See, e.g.,


Subquadratic Algorithms for 3SUM.  Ilya Baran, Erik D. Demaine, Mihai Patrascu.


However, it is not clear to me whether this leads to a practical improvement in real implementations.

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