## Abstract

The paper considers broadcasting protocols in radio networks with known topology that are efficient in both time and energy. The radio network is modelled as an undirected graph *G* = (*V*, *E*) where |*V*| = *n*. It is assumed that during execution of the communication task every node in *V* is allowed to transmit at most once. Under this assumption it is shown that any radio broadcast protocol requires \({D+\Omega(\sqrt{n-D})}\) transmission rounds, where *D* is the diameter of *G*. This lower bound is complemented with an efficient construction of a deterministic protocol that accomplishes broadcasting in \({D+O(\sqrt{n}\log n)}\) rounds. Moreover, if we allow each node to transmit at most *k* times, the lower bound \({D+\Omega((n-D)^{1/(2k)})}\) on the number of transmission rounds holds. We also provide a randomised protocol that accomplishes broadcasting in \({D+O(kn^{1/(k-2)}\log^2 n)}\) rounds. The paper concludes with a number of open problems in the area.

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## References

- 1
Ambühl, C.: An optimal bound for the MST algorithm to compute energy efficient broadcast trees in wireless networks. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming, pp. 1139–1150 (2005)

- 2
Ambühl, C., Erlebach, T., Mihalak, M., Nunkesser, M.: Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In: Proceedings of 9th International Workshop on Approximation Algorithms for Combinatorial Optimisation Problems (APPROX), pp. 3–14 (2006)

- 3
Alon, N., Bar-Noy, A., Linial, N., Peleg, D.: A lower bound for radio broadcast. J. Comput. Syst. Sci.

**43**, 290–298 (1991) - 4
Bar-Yehuda, R., Goldreich, O., Itai, A.: On the time complexity of broadcasting in radio networks: an exponential gap between determinism and randomization. In: Proceedings of the 5th Symposium on Principles of Distributed Computing (PODC), pp. 98–107 (1986)

- 5
Berenbrink, P., Cooper, C., Hu, Z.: Energy efficient randomised communication in unknown adhoc networks. In: Proceedings of the 19th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 250–259 (2007)

- 6
Chlamtac, I., Weinstein, O.: The wave expansion approach to broadcasting in multihop radio networks. IEEE Trans. Commun.

**39**, 426–433 (1991) - 7
Cicalese, F., Manne, F., Xin, Q.: Faster centralised communication in radio networks. In: Proceedings of the 17th International Symposium on Algorithms and Computation (ISAAC), pp. 339–348 (2006)

- 8
Clementi, A.E.F., Crescenzi, P., Penna, P., Rossi, G., Vocca, P.: On the complexity of computing minimum energy consumption broadcast subgraphs. In: Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 121–131 (2001)

- 9
Elkin, M., Kortsarz, G.: Improved broadcast schedule for radio networks. In: Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 222–231 (2005)

- 10
Flammini, M., Navarra, A., Klasing, R., Perennes, S.: Improved approximation results for the minimum energy broadcasting problem. In: Proceedings of the DIALM-POMC Workshop on Foundations of Mobile Computing, pp. 85–91 (2004)

- 11
Gaber, I., Mansour, Y.: Centralised broadcast in multihop radio networks. J. Algorithms

**46**(1), 1–20 (2003) - 12
Gąsieniec, L., Peleg, D., Xin, Q.: Faster communication in known topology radio networks. Distrib. Comput.

**19**(4), 289–300 (2007) - 13
Gąsieniec, L., Potapov, I., Xin, Q.: Time efficient centralised gossiping in radio networks. Theor. Comput. Sci.

**383**(1), 45–58 (2007) - 14
Goldberg, M., Spencer, T.: An efficient parallel algorithm that finds independent sets of guaranteed size. SIAM J. Discrete Math.

**6**(3), 443–459 (1993) - 15
Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica

**20**(4), 374–387 (1998) - 16
Guha, S., Khuller, S.: Improved methods for approximating node-weighted Steiner trees and connected dominating sets. Inf. Comput.

**150**, 57–74 (1999) - 17
Klasing, R., Navarra, A., Papadopoulos, A., Perennes, S.: Adaptive broadcast consumption (ABC), a new heuristic and new bounds for the minimum energy broadcast routing problem. Networking

**3042**, 866–877 (2004) - 18
Kowalski, D.R., Pelc, A.: Centralised deterministic broadcasting in undirected multi-hop radio networks. In: Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimisation Problems (APPROX), pp. 171–182 (2004)

- 19
Kowalski, D.R., Pelc, A.: Optimal deterministic broadcasting in known topology radio networks. Distrib. Comput.

**19**(3), 185–195 (2007) - 20
Mitzenmacher, M., Upfal, E.: Probability and Computing. Cambridge University Press, Cambridge (2005)

- 21
Navarra, A.: Tighter bounds for the minimum energy broadcasting problem. In: Proceedings of the 3rd International Symposium on Modeling and Optimisation in Mobile, Ad Hoc and Wireless Networks, pp. 313–322 (2005)

- 22
Turan, P.: On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok

**48**, 436–452 (1941) - 23
Wan, P.J., Calinescu, G., Li, X.Y., Frieder, O.: Minimum-energy broadcast routing in static ad hoc wireless networks. In: Proceedings of the 20th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pp. 1162–1171 (2001)

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## Additional information

The research of L. Gąsieniec, D.R. Kowalski and C. Su supported in part by the Royal Society grant *Algorithmic and Combinatorial Aspects of Radio Communication*, IJP - 2006/R2.

The research of E. Kantor and D. Peleg supported in part by grants from the Minerva Foundation and the Israel Ministry of Science.

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Gąsieniec, L., Kantor, E., Kowalski, D.R. *et al.* Time efficient k-shot broadcasting in known topology radio networks.
*Distrib. Comput.* **21, **117–127 (2008). https://doi.org/10.1007/s00446-008-0058-0

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### Keywords

- Span Tree
- Bipartite Graph
- Broadcast Message
- Ranking Scheme
- Unit Disk Graph