On Permuting Ability of a 2D Torus under XY routing

This paper introduces an exploration to permuting ability of a 2D torus under deterministic XY routing. The research is carried out for a number of communication models, namely, unidirectional uniaxial, bidirectional uniaxial, unidirectional biaxial, and bidirectional biaxial. Necessary and sufficient conditions of blocking occurrence in a 2D torus for uniaxial models are expressed mathematically with the use of congruence notion from number theory. Some examples of applying the technique are given. Frequently used permutations admissibility to a 2D torus of varied size under XY routing is checked computationally for aforementioned communication models. A simple way for realizing perfect shuffle and bit reversal permutations on 2D torus under XY routing is found.