The Remote Point Problem, Small Bias Space, and Expanding Generator Sets

TL;DR

This paper presents efficient parallel algorithms for the Remote Point Problem over various groups using epsilon-bias spaces, and explores their connection to constructing expanding generator sets.

Contribution

It extends the Remote Point Problem to arbitrary groups, providing NC^2 algorithms for Abelian groups and polynomial-time algorithms for nonabelian groups, linking to expanding generator set construction.

Findings

NC^2 algorithm for RPP over Abelian groups

Polynomial-time algorithm for RPP over nonabelian groups

Connection established between RPP solutions and expanding generator sets

Abstract

Using $\epsilon$-bias spaces over $F_2$, we show that the Remote Point Problem (RPP), introduced by Alon et al [APY09], has an $NC^2$ algorithm (achieving the same parameters as [APY09]). We study a generalization of the Remote Point Problem to groups: we replace $F^n$ by $G^n$ for an arbitrary fixed group $G$. When $G$ is Abelian, we give an $NC^2$ algorithm for RPP, again using $\epsilon$-bias spaces. For nonabelian $G$, we give a deterministic polynomial-time algorithm for RPP. We also show the connection to construction of expanding generator sets for the group $G^n$. All our algorithms for the RPP achieve essentially the same parameters as [APY09].