, Volume 50, Issue 4, pp 584-596
Date: 09 Sep 2007

Subquadratic Algorithms for 3SUM

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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.