Combinatiorial algorithms for wireless information flow

Abstract

A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr et al. [2007b] proposed a linear deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the adjacency matrices describing source-destination cuts.In this article, we develop a polynomial-time algorithm that discovers the relay encoding strategy to achieve the min-cut value in linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between channels to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple one-symbol processing at the intermediate nodes, thereby constructively yielding finite-length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network.

Improved combinatorial algorithms for wireless information flow

“…Table II lists the comparison results between different algorithms for finding the unicast capacity of linear deterministic wireless relay networks, specially in their complexity. O(M |E|C 5 ) Always higher than ours [4] O(d|Vx|C 5 + |Vy|C 5 ) especially when C is large [5] O(L 8 M 12 h 3 0 +LM 6 Ch 4 0 ) Always higher than ours, especially when M or L is large [6] O(L 1.5 M 3.5 log(M L)) or O(LM 3 log M )…”

Compress-and-Forward Scheme for Relay Networks: Backword Decoding and Connection to Bisubmodular Flows

“…Flows over linking network Random Linear Coding [2] Quantize-map-and-forward [2] Noisy network coding [1] Compress-and-forward Bisubmodular node flows The bisubmodular capacitated graph presented here is motivated by the ideas of linking systems and flows introduced in [6,7,8,9] in the context of the linear deterministic network. The linear deterministic network was introduced in [2] as a model that captures many features of the wireless network.…”

“…Random coding argument was used to show the existence of schemes that achieve capacity of the linear deterministic network [1,2]. On the other hand [6,7] developed a polynomial time algorithm that discovers the relay encoding strategy using a notion of linear independence between channels. Taking this concept forward, in [8,9], the concept of flow was introduced for the linear deterministic network.…”

On Relay Placement for Deterministic Line Networks

“…Such deterministic models can provide guidelines and inspire strategies for Gaussian networks. Efficient algorithms to find coding strategies for linear deterministic networks are known [4].…”