Reducing 3XOR to listing triangles, an exposition

The 3SUM problem asks if there are three integers a, b, c summing to 0 in a given set of n integers of magnitude poly(n). This problem can be easily solved in time Õ(n). (Throughout this note, Õ and Ω̃ hide subpolynomial factors n.) It seems natural to believe that this problem also requires time Ω̃(n), and this has been confirmed in some restricted models.[Eri99, AC05] The importance of this belief was brought to the forefront by Gajentaan and Overmars who show that the belief implies lower bounds for a number of problems in computational geometry;[GO95] and the list of such reductions has grown ever since. Recently, a series of exciting papers by Baran, Demaine, Pǎtraşcu, Vassilevska, and Williams set the stage for, and establish, reductions from 3SUM to new types of problems which are not defined in terms of summation.[BDP08, VW09, PW10, Pǎt10] In particular, Pǎtraşcu reduces 3SUM to the problem of listing triangles of a graph.[Pǎt10] In this note we present this reduction by Pǎtraşcu but for a variant of the 3SUM problem which we call 3XOR. The problem 3XOR is like 3SUM except that integer summation is replaced with bit-wise xor. So one can think of 3XOR as asking if a given n×O(lg n) matrix over the field with two elements has a linear combination of length 3. This problem is likely less relevant to computational geometry, but is otherwise quite natural. Similarly to 3SUM, 3XOR can be solved in time Õ(n), and it seems natural to conjecture that 3XOR requires time Ω̃(n). We now state the reduction we present.