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On minimizing the total power of k-strongly connected wireless networks

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Abstract

Given a wireless network, we want to assign each node a transmission power, which will enable transmission between any two nodes (via other nodes). Moreover, due to possible faults, we want to have at least k vertex-disjoint paths from any node to any other, where k is some fixed integer, depending on the reliability of the nodes. The goal is to achieve this directed k-strong connectivity with a minimal overall power assignment. The problem is NP-Hard for any k ≥ 1 already for planar networks. Here we first present an optimal power assignment for uniformly spaced nodes on a line for any k ≥ 1. We also prove a number of useful properties of power assignment which are also of independent interest. Based on it, we design an approximation algorithm for linear radio networks with factor \(\hbox{min}\left\{2,\left(\frac {\Updelta} {\delta}\right)^\alpha \right\},\) where Δ and δ are the maximal and minimal distances between adjacent nodes respectively and parameter α ≥ 1 being the distance-power gradient. We then extend it to the weighted version. Finally, we develop an approximation algorithm with factor O(k 2), for planar case, which is, to the best of our knowledge, the first non-trivial result for this problem.

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Notes

  1. A Hamiltonian cycle is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once. A graph possessing a Hamiltonian circuit is said to be a Hamiltonian graph or simply Hamiltonian.

  2. It is also easy to verify that r = 2α−1 for any α ≥ 2.

  3. Square of graph G = (V,E) is a graph G 2 = (V,E 2), where e = (u,v) ∈ E 2 if there is a path in G from u to v using at most 2 edges.

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Acknowledgements

The authors wish to thank Daniel Berend for long and fruitful discussions, valuable suggestions and a lot of help and support. The authors would also like to thank anonymous reviewers for their helpful comments that improved the paper. Hanan Shpungin has been supported in part by the Lynn and William Frankel Center for Computer Sciences. Work by Michael Segal has been supported by REMON (4G networking) consortium.

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Authors and Affiliations

  1. Computer Sciences Department, Ben Gurion University of the Negev, Beer-Sheva, Israel

    Hanan Shpungin

  2. Communication Systems Engineering Department, Ben Gurion University of the Negev, Beer-Sheva, Israel

    Michael Segal

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  1. Hanan Shpungin
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  2. Michael Segal
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Correspondence to Michael Segal.

Additional information

Preliminary version of this paper has been appeared at ACM/SIGMOBILE International Workshop on Foundation of Mobile Computing DIALM-POMC’05.

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Shpungin, H., Segal, M. On minimizing the total power of k-strongly connected wireless networks. Wireless Netw 16, 1075–1089 (2010). https://doi.org/10.1007/s11276-009-0189-7

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  • DOI: https://doi.org/10.1007/s11276-009-0189-7

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