Channel Assignment in Wireless Networks modeled as Cellular and Higher dimensional grids

This work was also with Prof. Shende and fellow students. Following the work done in our previous paper, we extended our work to find optimal and near-optimal channel assignment schemes to networks whose topologies resemble cellular grids, and d-dimensional square grids. We came up with the following results:

1. New algorithms were formulated for channel assignment in wireless networks modeled as cellular grids, with d₁ being in quadratic in s. The previous best known results for cellular grids have been restricted to s ≤ 4 and d₁ ≤ 2.
   
2. Optimal L(2,1) and L(2,1,1)-colorings were presented for square grids in all dimensions d > 0. We also introduced the notion of a coloring schema for the d-dimensional square grid, and an algorithm that, given a coloring schema, assigns colors to the grid satisfying the schema constraints.

Currently, we are trying to establish a tight upper bound for d₁ in networks modeled as cellular and square grids. We have found coloring schemes for certain networks (where s is odd i.e. s = 2k + 1) and d₁ = k². We conjecture that k² is indeed the upper bound on d₁ in such networks.