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

Using ε-bias spaces over F2, we show that the Remote Point Problem (RPP), introduced by Alon et al [APY09], has an NC algorithm (achieving the same parameters as [APY09]). We study a generalization of the Remote Point Problem to groups: we replace Fn 2 by Gn for an arbitrary fixed group G. When G is Abelian we give an NC algorithm for RPP, again using ε-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 Gn. All our algorithms for the RPP achieve essentially the same parameters as [APY09].