We consider the problem of distributed computation of a target function over a multiple-access channel. If the target and channel functions are matched (i.e., compute the same function), significant performance gains can be obtained by jointly designing the computation and communication tasks. However, in most situations there is mismatch between these two functions. In this work, we analyze the impact of this mismatch on the performance gains achievable with joint computation and communication designs over separation-based designs. We show that for most pairs of target and channel functions there is no such gain, and separation of computation and communication is optimal.
In most of these works, communication channels are represented as orthogonal point-to-point links. When the channel itself introduces signal interaction, as is the case for a MAC, there can be a benefit from jointly handling the communication and computation tasks as illustrated in @cite_9 . Function computation over MACs has been studied in @cite_12 @cite_8 @cite_17 @cite_7 and references therein.
@cite_7 @cite_8 @cite_9 @cite_12 @cite_17
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The following network computing problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e., the “computing capacity”. The network coding problem for a single-receiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network min-cut upper bound. We extend the definition of min-cut to the network computing problem and show that the min-cut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multi-edge tree networks. It is also tight for computing linear target functions in any network. We also study the bound's tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the min-cut bound. Traditionally, interference is considered harmful. Wireless networks strive to avoid scheduling multiple transmissions at the same time in order to prevent interference. This paper adopts the opposite approach; it encourages strategically picked senders to interfere. Instead of forwarding packets, routers forward the interfering signals. The destination leverages network-level information to cancel the interference and recover the signal destined to it. The result is analog network coding because it mixes signals not bits. So, what if wireless routers forward signals instead of packets? Theoretically, such an approach doubles the capacity of the canonical 2-way relay network. Surprisingly, it is also practical. We implement our design using software radios and show that it achieves significantly higher throughput than both traditional wireless routing and prior work on wireless network coding. Let (U_ i ,V_ i ) _ i=1 ^ n be a source of independent identically distributed (i.i.d.) discrete random variables with joint probability mass function p(u,v) and common part w=f(u)=g(v) in the sense of Witsenhausen, Gacs, and Korner. It is shown that such a source can be sent with arbitrarily small probability of error over a multiple access channel (MAC) X_ 1 X_ 2 , Y,p(y|x_ 1 ,x_ 2 ) , with allowed codes x_ l (u), x_ 2 (v) if there exist probability mass functions p(s), p(x_ 1 |s,u),p(x_ 2 |s,v) , such that H(U|V) H(V|U ) H(U,V|W) H(U,V) where p(s,u,v,x_ 1 ,x_ 2 ,y), Xl, X2, y)=p(s)p(u,v)p(x_ 1 |u,s)p(x_ 2 |v,s)p(y|x_ 1 ,x_ 2 ). lifts region includes the multiple access channel region and the Slepian-Wolf data compression region as special cases. A main distinguishing feature of a wireless network compared with a wired network is its broadcast nature, in which the signal transmitted by a node may reach several other nodes, and a node may receive signals from several other nodes simultaneously. Rather than a blessing, this feature is treated more as an interference-inducing nuisance in most wireless networks today (e.g., IEEE 802.11). The goal of this paper is to show how the concept of network coding can be applied at the physical layer to turn the broadcast property into a capacity-boosting advantage in wireless ad hoc networks. Specifically, we propose a physical-layer network coding (PNC) scheme to coordinate transmissions among nodes. In contrast to "straightforward" network coding which performs coding arithmetic on digital bit streams after they have been received, PNC makes use of the additive nature of simultaneously arriving electromagnetic (EM) waves for equivalent coding operation. PNC can yield higher capacity than straight-forward network coding when applied to wireless networks. We believe this is a first paper that ventures into EM-wave-based network coding at the physical layer and demonstrates its potential for boosting network capacity. PNC opens up a whole new research area because of its implications and new design requirements for the physical, MAC, and network layers of ad hoc wireless stations. The resolution of the many outstanding but interesting issues in PNC may lead to a revolutionary new paradigm for wireless ad hoc networking. The problem of reliably reconstructing a function of sources over a multiple-access channel (MAC) is considered. It is shown that there is no source-channel separation theorem even when the individual sources are independent. Joint source-channel strategies are developed that are optimal when the structure of the channel probability transition matrix and the function are appropriately matched. Even when the channel and function are mismatched, these computation codes often outperform separation-based strategies. Achievable distortions are given for the distributed refinement of the sum of Gaussian sources over a Gaussian multiple-access channel with a joint source-channel lattice code. Finally, computation codes are used to determine the multicast capacity of finite-field multiple-access networks, thus linking them to network coding.