Abstract
We consider the multihop broadcasting problem for n nodes placed uniformly at random in a disk and investigate the number of hops required to transmit a signal from the central node to all other nodes under three communication models: Unit-Disk-Graph (UDG), Signal-to-Noise-Ratio (SNR), and the wave superposition model of multiple input/multiple output (MIMO).
In the MIMO model, informed nodes cooperate to produce a stronger superposed signal. We do not consider the problem of transmitting a full message nor do we consider interference with other messages. In each round, the informed senders try to deliver to other nodes the required signal strength such that the received signal can be distinguished from the noise.
We assume a sufficiently high node density \(\rho = \varOmega (\log n)\) in order to launch the broadcasting process. In the unit-disk graph model, broadcasting takes \(\mathcal {O}(\sqrt{n/\rho })\) rounds. In the other models, we use an Expanding Disk Broadcasting Algorithm, where in a round only triggered nodes within a certain distance from the initiator node contribute to the broadcasting operation.
This algorithm achieves a broadcast in only \(\mathcal {O}\left( \frac{\log n}{\log \rho }\right) \) rounds in the SNR-model. Adapted to the MISO model, it broadcasts within \(\mathcal {O}(\log \log n - \log \log \rho )\) rounds. All bounds are asymptotically tight and hold with high probability, i.e. \(1- n^{-\mathcal {O}(1)}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Avin, C., Emek, Y., Kantor, E., Lotker, Z., Peleg, D., Roditty, L.: SINR diagrams: convexity and its applications in wireless networks. J. ACM 59(4), 18 (2012)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)
Dong, L., Petropulu, A., Poor, H.: A cross-layer approach to collaborative beamforming for wireless ad hoc networks. IEEE Trans. Signal Process. 56(7), 2981–2993 (2008)
Ferrari, F., Zimmerling, M., Thiele, L., Saukh, O.: Efficient network flooding and time synchronization with glossy. In: Proceedings of the 10th ACM/IEEE International Conference on Information Processing in Sensor Networks, pp. 73–84. IEEE, Chicago, April 2011
Ferrari, F., Zimmerling, M., Mottola, L., Thiele, L.: Low-power wireless bus. In: Proceedings of the 10th ACM Conference on Embedded Network Sensor Systems, SenSys 2012, pp. 1–14. ACM, New York (2012)
Franceschetti, M., Migliore, M.D., Minero, P.: The capacity of wireless networks: information-theoretic and physical limits. IEEE Trans. Inf. Theory 55(8), 3413–3424 (2009)
de Freitas, E.P., da Costa, J.P.C.L., de Almeida, A.L.F., Marinho, M.: Applying MIMO techniques to minimize energy consumption for long distances communications in wireless sensor networks. In: Andreev, S., Balandin, S., Koucheryavy, Y. (eds.) NEW2AN/ruSMART -2012. LNCS, vol. 7469, pp. 379–390. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32686-8_35
Gandhi, R., Mishra, A., Parthasarathy, S.: Minimizing broadcast latency and redundancy in ad hoc networks. IEEE/ACM Trans. Network. (TON) 16(4), 840–851 (2008)
Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Trans. Inf. Theory 46, 388–404 (2000)
Halldórsson, M.M., Tonoyan, T.: Leveraging indirect signaling for topology inference and fast broadcast. In: Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, pp. 85–93. ACM, New York (2018)
Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963)
Janson, T.: Energy-efficient collaborative beamforming in wireless ad hoc networks. Ph.D. thesis, University of Freiburg, Germany (2015)
Janson, T., Schindelhauer, C.: Analyzing randomly placed multiple antennas for MIMO wireless communication. In: Fifth International Workshop on Selected Topics in Mobile and Wireless Computing (IEEE STWiMob), Barcelona (2012)
Janson, T., Schindelhauer, C.: Broadcasting in logarithmic time for ad hoc network nodes on a line using MIMO. In: Proceedings of the 25th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2013. ACM, July 2013
Janson, T., Schindelhauer, C.: Ad-Hoc network unicast in O(log log n) using beamforming. http://arxiv.org/abs/1405.0417, May 2014
Janson, T., Schindelhauer, C.: Cooperative beamforming in ad-hoc networks with sublinear transmission power. In: IEEE 10th International Conference on Wireless and Mobile Computing. Networking and Communications (WiMob), pp. 144–151. IEEE, Larnaca, October 2014
Janson, T., Schindelhauer, C.: Self-synchronized cooperative beamforming in ad-hoc networks. In: 16th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2014), Paderborn, Germany, September 2014
Jeon, S.W., Chung, S.Y.: Two-phase opportunistic broadcasting in large wireless networks. In: IEEE International Symposium on Information Theory, ISIT 2007, pp. 2771–2775. IEEE (2007)
Kumberg, T., Schindelhauer, C., Reindl, L.: Exploiting concurrent wake-up transmissions using beat frequencies. Sensors 17(8), 1717 (2017)
Lebhar, E., Lotker, Z.: Unit disk graph and physical interference model: putting pieces together. In: IEEE International Symposium on Parallel Distributed Processing (IPDPS 2009), pp. 1–8, May 2009
Merzakreeva, A., Özgür, A., Lévêque, O.: Telescopic beamforming for large wireless networks. In: IEEE International Symposium on Information Theory, Istanbul (2013)
Niesen, U., Gupta, P., Shah, D.: On capacity scaling in arbitrary wireless networks. IEEE Trans. Inf. Theory 55(9), 3959–3982 (2009)
Oak, A.: Analysis of a collaborative iterative MISO broadcasting algorithm. Master’s thesis, University of Freiburg, Freiburg, Germany, March 2018
Oyman, O., Paulraj, A.J.: Power-bandwidth tradeoff in dense multi-antenna relay networks. IEEE Trans. Wireless Commun. 6(6) (2007)
Oyman, O., Paulraj, A.J.: Cooperative OFDMA and distributed MIMO relaying over dense wireless networks, uS Patent 8,027,301, 27 September 2011
Ozgur, A., Lévêque, O., David, N.: Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks. IEEE Trans. Inf. Theory 53(10), 3549–3572 (2007)
Özgür, A., Leveque, O., Tse, D.: Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks. IEEE Trans. Inf. Theory 53(10), 3549–3572 (2007)
Özgür, A., Lévêque, O., Tse, D.: Spatial degrees of freedom of large distributed mimo systems and wireless ad hoc networks. IEEE J. Sel. Areas Commun. 31(EPFL–ARTICLE–185421), 202–214 (2013)
Peleg, D.: Time-efficient broadcasting in radio networks: a review. In: Janowski, T., Mohanty, H. (eds.) ICDCIT 2007. LNCS, vol. 4882, pp. 1–18. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-77115-9_1
Schindelhauer, C., Oak, A., Janson, T.: Collaborative broadcast in O(log log n) rounds. arXiv e-prints arXiv:1906.05153, June 2019
Sirkeci-Mergen, B., Gastpar, M.C.: On the broadcast capacity of wireless networks with cooperative relays. IEEE Trans. Inf. Theory 56(8), 3847–3861 (2010)
Sirkeci-Mergen, B., Scaglione, A., Mergen, G.: Asymptotic analysis of multistage cooperative broadcast in wireless networks. IEEE/ACM Trans. Netw. 14(SI), 2531–2550 (2006)
Sutton, F., Buchli, B., Beutel, J., Thiele, L.: Zippy: on-demand network flooding. In: Proceedings of the 13th ACM Conference on Embedded Networked Sensor Systems, pp. 45–58. ACM (2015)
Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press, New York (2005)
Xue, F., Kumar, P.R.: The number of neighbors needed for connectivity of wireless networks. Wireless Netw. 10(2), 169–181 (2004)
Acknowledgments
We like to thank the organizers of the Dagstuhl Seminar 17271, July 2–7, 2017, Foundations of Wireless Networking, where this research has begun and first results have been found. We would like to thank Alexander Leibold, who performed and checked the automated proofs and anonymous reviewers of a previous version for their detailed and valuable input. We would also like to thank Tigran Tonoyan, Magnús M. Halldórsson and Zvi Lotker for many fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Schindelhauer, C., Oak, A., Janson, T. (2019). Collaborative Broadcast in \(\mathcal {O}(\log \log n)\) Rounds. In: Dressler, F., Scheideler, C. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2019. Lecture Notes in Computer Science(), vol 11931. Springer, Cham. https://doi.org/10.1007/978-3-030-34405-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-34405-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-34404-7
Online ISBN: 978-3-030-34405-4
eBook Packages: Computer ScienceComputer Science (R0)