The Strong 3SUM-INDEXING Conjecture is False

TL;DR

This paper disproves the Strong 3SUM-INDEXING Conjecture by reducing the problem to function inversion and applying a known algorithm, showing more efficient solutions are possible.

Contribution

It demonstrates that the longstanding conjecture about the hardness of 3SUM-Indexing is false through a novel reduction and application of existing function inversion algorithms.

Findings

The conjecture that 3SUM-Indexing cannot be solved with certain space and time bounds is false.

A reduction from 3SUM-Indexing to function inversion is established.

An existing function inversion algorithm can be used to solve 3SUM-Indexing efficiently.

Abstract

In the 3SUM-Indexing problem the goal is to preprocess two lists of elements from $U$, $A=(a_1,a_2,\ldots,a_n)$ and $B=(b_1,b_2,...,b_n)$, such that given an element $c\in U$ one can quickly determine whether there exists a pair $(a,b)\in A \times B$ where $a+b=c$. Goldstein et al.~[WADS'2017] conjectured that there is no algorithm for 3SUM-Indexing which uses $n^{2-\Omega(1)}$ space and $n^{1-\Omega(1)}$ query time. We show that the conjecture is false by reducing the 3SUM-Indexing problem to the problem of inverting functions, and then applying an algorithm of Fiat and Naor [SICOMP'1999] for inverting functions.