Efficient algorithm for linear diophantine equations in two variables

TL;DR

This paper introduces an optimized recursive algorithm, DEA-OPTD, for solving linear diophantine equations in two variables, which reduces recursive calls and outperforms existing algorithms on most inputs.

Contribution

The paper presents DEA-OPTD, an improved version of DEA-R, with more efficient recursive computations and theoretical analysis showing its potential advantages over previous methods.

Findings

DEA-OPTD reduces recursive calls compared to DEA-R.

DEA-OPTDI outperforms other algorithms on at least 96% of inputs.

Theoretical bounds suggest when DEA-OPTD is more efficient.

Abstract

Solving linear diophantine equations in two variables have applications in computer science and mathematics. In this paper, we revisit an algorithm for solving linear diophantine equations in two variables, which we refer as DEA-R algorithm. The DEA-R algorithm always incurs equal or less number of recursions or recursive calls as compared to extended euclidean algorithm. With the objective of taking advantage of the less number of recursive calls , we propose an optimized version of the DEA-R algorithm as DEA-OPTD. In the recursive function calls in DEA-OPTD, we propose a sequence of more efficient computations. We do a theoretical comparison of the execution times of DEA-OPTD algorithm and DEA-R algorithm to find any possible bound on the value of $c$ for DEA-OPTD being better than DEA-R. We implement and compare an iterative version of DEA-OPTD (DEA-OPTDI) with two versions of a widely used algorithm on an specific input setting. In this comparison, we find out that our algorithm outperforms on the other algorithm against atleast 96% of the inputs.