Abstract
We give an algorithm for constructing a connected spanning subgraphs(panner) of a wireless network modelled as a unit disk graph with nodes of irregular transmission ranges, whereby for some parameter 0 < r ≤ 1 the transmission range of a node includes the entire disk around the node of radius at least r and it does not include any node at distance more than one. The construction of a spanner is distributed and local in the sense that nodes use only information at their vicinity, moreover for a given integer k ≥ 2 each node needs only consider all the nodes at distance at most k hops from it. The resulting spanner has maximum degree at most 3 + \(\frac{6}{\pi r}\) + \(\frac{r+1}{r^{2}}\), when 0 < r <1 (and at most five, when r = 1). Furthermore it is shown that the spanner is planar provided that the distance between any two nodes is at least \(\sqrt{1-r^{2}}\). If the spanner is planar then for k ≥ 2 the sum of the Euclidean lengths of the edges of the spanner is at most \(\frac{kr+1}{kr-1}\) times the sum of the Euclidean lengths of the edges of a minimum weight Euclidean spanning tree.
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Chávez, E., Dobrev, S., Kranakis, E., Opatrny, J., Stacho, L., Urrutia, J. (2006). Local Construction of Planar Spanners in Unit Disk Graphs with Irregular Transmission Ranges. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_29
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DOI: https://doi.org/10.1007/11682462_29
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