Abstract
We obtain subquadratic algorithms for 3SUM on integers and rationals in several models. On a standard word RAM with w-bit words, we obtain a running time of \(O(n^{2}/\max\{\frac{w}{\lg^{2}w},\frac{\lg^{2}n}{(\lg\lg n)^{2}}\})\) . In the circuit RAM with one nonstandard AC 0 operation, we obtain \(O(n^{2}/\frac{w^{2}}{\lg^{2}w})\) . In external memory, we achieve O(n 2/(MB)), even under the standard assumption of data indivisibility. Cache-obliviously, we obtain a running time of \(O(n^{2}/\frac{MB}{\lg^{2}M})\) . In all cases, our speedup is almost quadratic in the “parallelism” the model can afford, which may be the best possible. Our algorithms are Las Vegas randomized; time bounds hold in expectation, and in most cases, with high probability.
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Baran, I., Demaine, E.D. & Pǎtraşcu, M. Subquadratic Algorithms for 3SUM. Algorithmica 50, 584–596 (2008). https://doi.org/10.1007/s00453-007-9036-3
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DOI: https://doi.org/10.1007/s00453-007-9036-3