## Abstract

Efficient communication in wireless networks is typically challenged by the possibility of interference among several transmitting nodes. Much important research has been invested in decreasing the number of collisions in order to obtain faster algorithms for communication in such networks. This paper proposes a novel approach for wireless communication, which embraces collisions rather than avoiding them, over an additive channel. It introduces a coding technique called *Bounded-Contention Coding* (*BCC*) that allows collisions to be successfully decoded by the receiving nodes into the original transmissions and whose complexity depends on a bound on the contention among the transmitters. BCC enables *deterministic* local broadcast in a network with \(n\) nodes and at most \(a\) transmitters with information of \(\ell \) bits each within \(O(a\log {n}+a\ell )\) bits of communication with full-duplex radios, and \(O((a\log {n}+a\ell )(\log {n}))\) bits, with high probability, with half-duplex radios. When combined with random linear network coding, BCC gives *global* broadcast within \(O((D+a+\log {n})(a\log {n}+\ell ))\) bits, with high probability. This also holds in dynamic networks that can change arbitrarily over time by a worst-case adversary. When no bound on the contention is given, it is shown how to probabilistically estimate it and obtain global broadcast that is adaptive to the true contention in the network.

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## Notes

- 1.
Throughout this paper, high probability refers to values that are at least \(1-1/n^c\) for some constant \(c \ge 1\).

- 2.
For every set \(S\subseteq C\), let \(\bar{y}_S\) be the length \(M\) vector characterizing the set \(S\), i.e., \(\bar{y}_S(i)=1\) if and only if the \(i\)-th column is an element in \(S\). Consider the multiplication of the matrix \(H\) by the vector \(\bar{y}_S\). If \(H\cdot \bar{y}_S^T=0\) then \(\bar{y}\) is orthogonal to \(D^{\bot }\), hence \(\bar{y} \in (D^{\bot })^{\bot }=D\). This implies that the Hamming weight of \(\bar{y}\) is at least \(2a+1\).

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## Acknowledgments

The authors thank Seth Gilbert for useful discussions regarding probabilistic retransmissions, MinJi Kim and Ali ParandehGheibi for many discussions about the XOR collisions model addressed in this paper, and Amir Shpilka for pointing out the existence of the simple codes we use to implement our BCC framework. We further thank the anonymous reviewers for their effort and valuable feedback which improved and clarified the presentation of this work in many ways. This work was supported in part by the Simons Postdoctoral Fellows Program, ISF Grant 1696/14, AFOSR Contract Number: FA9550-13-1-0042, AFOSR Contract Number: FA9550-14-1-0403, and NSF Award 0939370-CCF.

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## Additional information

A preliminary version of this paper appeared in Proceedings of the 26th International Symposium on Distributed Computing (DISC), pp. 91–105, 2012.

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### Cite this article

Censor-Hillel, K., Haeupler, B., Lynch, N. *et al.* Bounded-Contention Coding for the additive network model.
*Distrib. Comput.* **28, **297–308 (2015). https://doi.org/10.1007/s00446-015-0244-9

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### Keywords

- Wireless networks
- High SNR
- Coding
- Additive channel