Formula omitted] labeling of d-dimensional grids

Abstract

In this paper, we address the problem of λ labelings, that was introduced in the context of frequency assignment for telecommunication networks. In this model, stations within a given radius r must use frequencies that differ at least by a value p, while stations that are within a larger radius r ′ > r must use frequencies that differ by at least another value q. The aim is to minimize the span of frequencies used in the network. This can be modeled by a graph coloring problem, called the L ( p , q ) labeling, where one wants to label vertices of the graph G modeling the network by integers in the range [ 0 ; M ] , in such a way that: (1) neighbors in G are assigned colors differing by at least p and (2) vertices at distance 2 in G are assigned colors differing by at least q, while minimizing the value of M. M is then called the λ number of G, and is denoted by λ q p ( G ) . In this paper, we study the L ( p , q ) labeling for a specific class of networks, namely the d-dimensional grid G d = G [ n 1 , n 2 , … , n d ] . We give bounds on the value of the λ number of an L ( p , q ) labeling for any d ⩾ 1 and p , q ⩾ 0 . Some of these results are optimal (namely, in the following cases: (1) p = 0 , (2) q = 0 , (3) q = 1 (4) p , q ⩾ 1 , p = α · q with 1 ⩽ α ⩽ 2 d and (5) p ⩾ 2 dq + 1 ); when the results we obtain are not optimal, we observe that the bounds differ by an additive factor never exceeding 2 q - 2 . The optimal result we obtain in the case q = 1 answers an open problem stated by Dubhashi et al. [Channel assignment for wireless networks modelled as d -dimensional square grids, in: Proceedings of the IWDC’02, International Workshop on Distributed Computing, Lecture Notes in Computer Science, vol. 2571, Springer, Berlin, 2002, pp. 130–141], and generalizes results from [A.A Bertossi, C.M. Pinotti, R.B. Tan, Efficient use of radio spectrum in wireless networks with channel separation between close stations, in: Proceedings of the DIAL M for Mobility 2000, Fourth International Workshop on Discrete Algorithms and Methods for Mobile Computing and communications, 2000; A. Dubhashi, S. MVS, A. Pati, S. R., A.M. Shende, Channel assignment for wireless networks modelled as d -dimensional square grids, in: Proceedings of the IWDC’02, International Workshop on Distributed Computing, Lecture Notes in Computer Science, vol. 2571, Springer, Berlin, 2002, pp. 130–141]. We also apply our results to get upper bounds for the L ( p , q ) labeling of d-dimensional hypercubes.