More Logarithmic-Factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems

摘要

We present an algorithm that solves the 3SUM problem for n real numbers in O((n~2/log~2 n)(log log n)~(O(1))) time, improving previous solutions by about a logarithmic factor. Our framework for shaving off two logarithmic factors can be applied to other problems, such as (median,+)-convolution/matrix multiplication and algebraic generalizations of 3SUM. We also obtain the first subquadratic results on some 3SUM-hard problems in computational geometry, for example, deciding whether (the interiors of) a constant number of simple polygons have a common intersection.