A new algorithm for solving the rSUM problem

TL;DR

This paper introduces a novel algorithm for the rSUM problem that operates efficiently by analyzing binary bits of integers, achieving sub-quadratic time complexity in some cases and using memory proportional to n log^3(n).

Contribution

It presents a new binary-based approach to solve the rSUM problem with improved time and memory efficiency, extending applicability to any natural r.

Findings

Achieves sub-quadratic time complexity in certain cases.

Uses memory on the order of n log^3(n).

Introduces a binary bit-based method to discard non-solution candidates.

Abstract

A determined algorithm is presented for solving the rSUM problem for any natural r with a sub-quadratic assessment of time complexity in some cases. In terms of an amount of memory used the obtained algorithm is the nlog^3(n) order. The idea of the obtained algorithm is based not considering integer numbers, but rather k (is a natural) successive bits of these numbers in the binary numeration system. It is shown that if a sum of integer numbers is equal to zero, then the sum of numbers presented by any k successive bits of these numbers must be sufficiently "close" to zero. This makes it possible to discard the numbers, which a fortiori, do not establish the solution.