## Abstract

In this paper we study the problem of building a constant-degree connected dominating set (CCDS), a network structure that can be used as a communication backbone, in the dual graph radio network model (Clementi et al. in J Parallel Distrib Comput 64:89–96, 2004; Kuhn et al. in Proceedings of the international symposium on principles of distributed computing 2009, Distrib Comput 24(3–4):187–206 2011, Proceedings of the international symposium on principles of distributed computing 2010). This model includes two types of links: *reliable*, which always deliver messages, and *unreliable*, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process \(u\) is provided with a local *link detector* set consisting of every neighbor connected to \(u\) by a reliable link. The algorithm solves the CCDS problem in \(O\left( \frac{\varDelta \log ^2{n}}{b} + \log ^3{n}\right) \) rounds, with high probability, where \(\varDelta \) is the maximum degree in the reliable link graph, \(n\) is the network size, and \(b\) is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in \(\log ^3{n}\) time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow-up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in \(O(\varDelta \)polylog\((n))\) time under the assumption of \(O(1)\) unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process’s local link detector set is sufficient to require \(\varOmega (\varDelta )\) rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time.

## Keywords

Radio networks Maximal independent set Connected dominating set Link detector## Notes

### Acknowledgments

The authors thank the anonymous reviewers for useful comments and suggestions that helped improve the presentation of this paper.

## References

- 1.Personal communication with Johannes Schneider, ETH Zurich (2011)Google Scholar
- 2.Abusubaih, M.: A new approach for interference measurement in 802.11 WLANs. In: Proceedings of the International Symposium on Personal Indoor and Mobile Radio Communications, pp. 2336–2341 (2010)Google Scholar
- 3.Aguayo, D., Bicket, J., Biswas, S., Morris, R., Chambers, B., De Couto, D.: MIT roofnet. In: Proceedings of the International Conference on Mobile Computing and Networking (2003)Google Scholar
- 4.Chandra, T.D., Toueg, S.: Unreliable failure detectors for reliable distributed systems. J. ACM
**43**(2), 225–267 (1996)CrossRefzbMATHMathSciNetGoogle Scholar - 5.Clementi, A., Monti, A., Silvestri, R.: Round robin is optimal for fault-tolerant broadcasting on wireless networks. J. Parallel Distrib. Comput.
**64**, 89–96 (2004)CrossRefzbMATHGoogle Scholar - 6.De Couto, D., Aguayo, D., Bicket, J., Morris, R.: A high-throughput path metric for multi-hop wireless routing. Wirel. Netw.
**11**(4), 419–434 (2005)CrossRefGoogle Scholar - 7.De Couto, D., Aguayo, D., Chambers, B., Morris, R.: Performance of multihop wireless networks: shortest path is not enough. ACM SIGCOMM Comput. Commun. Rev.
**33**(1), 83–88 (2003)CrossRefGoogle Scholar - 8.Ghaffari, M., Haeupler, B., Lynch, N., Newport, C.: Bounds on contention management in radio networks. In: Proceedings of the International Symposium on Distributed Computing, pp. 223–237 (2012)Google Scholar
- 9.Kim, K., Shin, K.: On accurate measurement of link quality in multi-hop wireless mesh networks. In: Proceedings of the Annual International Conference on Mobile Computing and Networking, pp. 38–49 (2006)Google Scholar
- 10.Kuhn, F.: The price of locality: exploring the complexity of distributed coordination primitives. Ph.D. Thesis, ETH Zurich (2005)Google Scholar
- 11.Kuhn, F., Lynch, N., Newport, C.: Brief announcement: hardness of broadcasting in wireless networks with unreliable communication. In: Proceedings of the International Symposium on Principles of Distributed Computing, pp. 330–331 (2009)Google Scholar
- 12.Kuhn, F., Lynch, N., Newport, C.: The abstract MAC layer. Distrib. Comput.
**24**(3–4), 187–206 (2011)CrossRefzbMATHGoogle Scholar - 13.Kuhn, F., Lynch, N., Newport, C., Oshman, R., Richa, A.: Broadcasting in unreliable radio networks. In: Proceedings of the International Symposium on Principles of Distributed Computing, pp. 336–345 (2010)Google Scholar
- 14.Kuhn, F., Moscibroda, T., Wattenhofer, R.: Initializing newly deployed ad hoc and sensor networks. In: Proceedings of the Annual International Conference on Mobile Computing and Networking, pp. 260–274 (2004) Google Scholar
- 15.Kuhn, F., Wattenhofer, R.: Constant-time distributed dominating set approximation. Distrib. Comput.
**17**(4), 303–310 (2005)CrossRefzbMATHGoogle Scholar - 16.Kuhn, F., Wattenhofer, R., Zollinger, A.: Ad hoc networks beyond unit disk graphs. Wirel. Netw.
**14**(5), 715–729 (2008)CrossRefGoogle Scholar - 17.Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput.
**15**(4), 1036–1053 (1986)CrossRefzbMATHMathSciNetGoogle Scholar - 18.Moscibroda, T., Wattenhofer, R.: Maximal independent sets in radio networks. In: Proceedings of the International Symposium on Principles of Distributed Computing, pp. 148–157 (2005)Google Scholar
- 19.Parthasarathy, S., Gandhi, R.: Distributed algorithms for coloring and domination in wireless ad hoc networks. In: Proceedings of the Conference on the Foundations of Software Technology and Theoretical Computer Science, pp. 447–459 (2005)Google Scholar
- 20.Ramachandran, K., Sheriff, I., Belding, E., Almeroth, K.: Routing Stability in Static Wireless Mesh Networks. In: Proceedings of the International Conference on Passive and Active Network Measurement, pp. 73–82 (2007)Google Scholar
- 21.Srinivasan, K., Kazandjieva, M., Agarwal, S., Levis, P.: The \(\beta \)-factor: measuring wireless link burstiness. In: Proceedings of the Conference on Embedded Networked Sensor System, pp. 29–42 (2008)Google Scholar
- 22.Wan, P., Alzoubi, K., Frieder, O.: Distributed construction of connected dominating sets in wireless ad hoc networks. In: Proceedings of the IEEE Conference on Computer Communications, pp. 1597–1604 (2002)Google Scholar
- 23.Yarvis, M., Conner, W., Krishnamurthy, L., Chhabra, J., Elliott, B., Mainwaring, A.: Real-world experiences with an interactive ad hoc sensor network. In: Proceedings of the International Conference of Parallel Processing, pp. 143–151 (2002)Google Scholar