A fast algorithm for the Frobenius problem in three variables

TL;DR

This paper introduces a new, faster algorithm for solving the Frobenius problem with three variables, improving computational efficiency while building on previous methods by Greenberg and Tripathi.

Contribution

The paper presents a novel algorithm for the three-variable Frobenius problem with a logarithmic worst-case complexity, differing significantly from Tripathi's approach.

Findings

Algorithm has logarithmic worst-case time complexity

Significantly different from Tripathi's algorithm

Improves computational efficiency for the Frobenius problem

Abstract

Given a set of three positive integers {a1, a2, a3}, denoted A, the Frobenius problem in three variables is to find the greatest integer which cannot be expressed in the following form, where x1, x2 and x3 are non-negative integers: x1*a1 + x2*a2 + x3*a3 The fastest known algorithm for solving the three variable case of the Frobenius problem was invented by H. Greenberg in 1988 whose worst case time complexity is a logarithmic function of A. In 2017 A. Tripathi presented another algorithm for solving the same problem. This article presents an algorithm whose foundation is the same as Tripathi's. However, the algorithm presented here is significantly different from Tripathi's and we show that its worst case time complexity also is a logarithmic function of A