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 .

In this paper, we study the L(p,q) labeling for a specific class of networks, namely the d-dimensional grid G_d=G[n₁,n₂,…,n_d]. We give bounds on the value of the λ number of an L(p,q) labeling for any d1 and p,q0. Some of these results are optimal (namely, in the following cases: (1) p=0, (2) q=0, (3) q=1 (4) p,q1, p=α·q with 1α2d and (5) p2dq+1); when the results we obtain are not optimal, we observe that the bounds differ by an additive factor never exceeding 2q-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.

Keywords: L(p,Q) labeling; Grids

Article Outline

1. Introduction

2. L(p,q) labeling of G_d

2.1. L(p,q) labeling when p=0 or q=0

2.2. L(p,q) labeling when p,q1

3. Conclusion

References

Part of this work was done while both authors were visiting the Universitat Politècnica de Catalunya in Barcelona. This visit was supported by the Egide Spanish–French exchange program PICASSO Number 04295SG.