On the problem of tracking mobile users in wireless communications networks

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The tracking strategy for mobile users in wireless communication networks proposed by Bar-Noy and Kessler in [ I ] essentiallyfinds a minimal set of reporting centers in the cellular network. The size of the set of reporting centers in the mobility graph, and the vicinity (i.e., the number of non reporting centers in the neighborhood of a reporting center} are measures of the cost of the two main operations, update and find, in the tracking problem. For arbitrary mobility graphs and for any vicinity value greater than I , the problem has been shown to be NP-complete. Optimal and near optimal solutions for special cases of the mobility graphs, including rings, grids, trees, have been discussed in [l]. In this note, we propose optimal greedy algorithms to solve the reporting center problem for vicinity of size 2 in interval graphs and proper interval graphs. The heuristic that guides our greedy solutions is to make a vertex a reporting center if and only if it has already a non reporting center among its neighbors. The proposed algorithms run in linear time.