Abstract :
In this paper, we address the problem of image 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 image 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 image labeling, where one wants to label vertices of the graph G modeling the network by integers in the range image, 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 image number of G, and is denoted by image.
In this paper, we study the image labeling for a specific class of networks, namely the d-dimensional grid image. We give bounds on the value of the image number of an image labeling for any image and image. Some of these results are optimal (namely, in the following cases: (1) image, (2) image, (3) image (4) image, image with image and (5) image); when the results we obtain are not optimal, we observe that the bounds differ by an additive factor never exceeding image. The optimal result we obtain in the case image answers an open problem stated by Dubhashi et al. [Channel assignment for wireless networks modelled as image-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 image 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 image-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 image labeling of d-dimensional hypercubes.
