A Subquadratic Algorithm for 3XOR

TL;DR

This paper introduces a simple deterministic quadratic-time algorithm for the 3XOR problem using Patricia tries, and also presents a randomized algorithm with expected subquadratic time, advancing the state of the art in this fundamental problem.

Contribution

The paper provides the first simple deterministic quadratic algorithm for 3XOR and a new randomized expected subquadratic algorithm, improving over previous methods.

Findings

Deterministic algorithm runs in quadratic time using Patricia trie structure.

Randomized algorithm achieves expected running time subquadratic in n.

Reductions show 3XOR's relation to other problems like SetDisjointness and SetIntersection.

Abstract

Given a set $X$ of $n$ binary words of equal length $w$, the 3XOR problem asks for three elements $a, b, c \in X$ such that $a \oplus b=c$, where $ \oplus$ denotes the bitwise XOR operation. The problem can be easily solved on a word RAM with word length $w$ in time $O(n^2 \log{n})$. Using Han's fast integer sorting algorithm (2002/2004) this can be reduced to $O(n^2 \log{\log{n}})$. With randomization or a sophisticated deterministic dictionary construction, creating a hash table for $X$ with constant lookup time leads to an algorithm with (expected) running time $O(n^2)$. At present, seemingly no faster algorithms are known. We present a surprisingly simple deterministic, quadratic time algorithm for 3XOR. Its core is a version of the Patricia trie for $X$, which makes it possible to traverse the set $a \oplus X$ in ascending order for arbitrary $a\in \{0, 1\}^{w}$ in linear time. Furthermore, we describe a randomized algorithm for 3XOR with expected running time $O(n^2\cdot\min\{\log^3{w}/w, (\log\log{n})^2/\log^2 n\})$. The algorithm transfers techniques to our setting that were used by Baran, Demaine, and P\u{a}tra\c{s}cu (2005/2008) for solving the related int3SUM problem (the same problem with integer addition in place of binary XOR) in expected time $o(n^2)$. As suggested by Jafargholi and Viola (2016), linear hash functions are employed. The latter authors also showed that assuming 3XOR needs expected running time $n^{2-o(1)}$ one can prove conditional lower bounds for triangle enumeration just as with 3SUM. We demonstrate that 3XOR can be reduced to other problems as well, treating the examples offline SetDisjointness and offline SetIntersection, which were studied for 3SUM by Kopelowitz, Pettie, and Porat (2016).