Computing efficiently the weighted greatest common divisor

TL;DR

This paper explores properties and optimization techniques for computing the weighted greatest common divisor (wgcd), introducing an ordering method that simplifies calculations and proves a new formula relating wgcd to gcd of tail elements.

Contribution

It presents a novel approach to efficiently compute wgcd by ordering weights and establishing a new formula linking wgcd to gcd of subsequences.

Findings

Ordering weights reduces computational complexity.

The paper proves a formula relating wgcd to gcd of tail elements.

Significantly decreases the numbers involved in wgcd calculations.

Abstract

In this paper we included some basic properties for weighted greatest common divisors, and discuss how to speed up computing the weighted greatest common divisor. By ordering the 'weights' we are able to significantly shorten the operations to computing wgcd. In the absence of an efficient algorithm for computing wgcd by ordering the weights, and using $\gcd$, we significantly reduce the numbers for which we want to compute wgcd. As a final result in this paper we prove that: If $\mathbf{x} = (x_{0},\dots ,x_{n})\in \mathbb{Z}^{n+1}$, with weights $\mathfrak{w}=(q_{0},\dots ,q_{n})$ and $q_{0}\leq \cdots \leq q_{n}$, then ${\rm wgcd}_w(\mathbf{x}) = {\rm wgcd}_w(y_0,y_1,\dots,y_n)$, where $y_i = \gcd(x_i,\dots,x_n)$, and $y_0\leq y_1 \leq\dots \leq y_n$.