Real Number Channel Assignments for Lattices

We consider the problem of assigning a numerical channel to each transmitter in a large regular array such that multiple levels of interference, which depend on the distance between transmitters, are avoided by sufficiently separating the channels. The goal is to find assignments that minimize the span of the labels used. A previous paper of the authors introduced a model for this problem using real number labellings of (possibly infinite) graphs G. Given reals k1, k2, . . . , kp ≥ 0, one denotes by λ(G; k1, k2, · · · , kp) the infimum of the spans of the labellings f of the vertices v of G, such that for any two vertices v and w, the difference in their labels is at least ki, where i is the distance between v and w in G. When p = 2, it is enough to determine λ(G; k, 1) for reals k ≥ 0; for G of bounded maximum degree, this will be a continuous, piecewise linear function of k. Here we consider this function for infinite regular lattices that model large planar networks, building on earlier efforts by other researchers. For the triangular lattice, we determine the function for k ≥ 1, which had previously been found for rational k ≥ 3 by Calamoneri. We also give bounds for 0 ≤ k ≤ 1. For the square lattice and the hexagonal lattice, we completely determine the function for k ≥ 0, which had been given for rational k ≥ 3 and k ≥ 2, respectively, by Calamoneri. Portions of it have been obtained by other researchers for infinite regular lattices that model large planar networks. Here we present the complete function λ(G; k, 1), for k ≥ 1 when G is the triangular, square, or hexagonal lattice. ∗Research supported in part by NSF grants DMS-0072187 and DMS-0302307. The principal results here were announced in an extended abstract for the International Workshop on Wireless, Mobile, and Ad Hoc Networks in the Proceedings of the 2005 IEEE International Parallel and Distributed Computing Symposium [16]. This research was also described in part in the second author’s dissertation [24].