Network Coding for Computing: Cut-Set Bounds

The following \textit{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 \textit{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 \textit{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 and 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.Comment: Submitted to the IEEE Transactions on Information Theory (Special Issue on Facets of Coding Theory: from Algorithms to Networks); Revised on Aug 9, 201

Zero-Error Coding for Computing with Encoder Side-Information

We study the zero-error source coding problem in which an encoder with Side Information (SI) $g(Y)$ transmits source symbols $X$ to a decoder. The decoder has SI $Y$ and wants to recover $f(X,Y)$ where $f,g$ are deterministic. We exhibit a condition on the source distribution and $g$ that we call "pairwise shared side information", such that the optimal rate has a single-letter expression. This condition is satisfied if every pair of source symbols "share" at least one SI symbol for all output of $g$. It has a practical interpretation, as $Y$ models a request made by the encoder on an image $X$, and $g(Y)$ corresponds to the type of request. It also has a graph-theoretical interpretation: under "pairwise shared side information" the characteristic graph can be written as a disjoint union of OR products. In the case where the source distribution is full-support, we provide an analytic expression for the optimal rate. We develop an example under "pairwise shared side information", and we show that the optimal coding scheme outperforms several strategies from the literature

Hypergraph-based Source Codes for Function Computation Under Maximal Distortion

This work investigates functional source coding problems with maximal distortion, motivated by approximate function computation in many modern applications. The maximal distortion treats imprecise reconstruction of a function value as good as perfect computation if it deviates less than a tolerance level, while treating reconstruction that differs by more than that level as a failure. Using a geometric understanding of the maximal distortion, we propose a hypergraph-based source coding scheme for function computation that is constructive in the sense that it gives an explicit procedure for defining auxiliary random variables. Moreover, we find that the hypergraph-based coding scheme achieves the optimal rate-distortion function in the setting of coding for computing with side information and the Berger-Tung sum-rate inner bound in the setting of distributed source coding for computing. It also achieves the El Gamal-Cover inner bound for multiple description coding for computing and is optimal for successive refinement and cascade multiple description problems for computing. Lastly, the benefit of complexity reduction of finding a forward test channel is shown for a class of Markov sources

Computation Over Gaussian Networks With Orthogonal Components

Function computation of arbitrarily correlated discrete sources over Gaussian networks with orthogonal components is studied. Two classes of functions are considered: the arithmetic sum function and the type function. The arithmetic sum function in this paper is defined as a set of multiple weighted arithmetic sums, which includes averaging of the sources and estimating each of the sources as special cases. The type or frequency histogram function counts the number of occurrences of each argument, which yields many important statistics such as mean, variance, maximum, minimum, median, and so on. The proposed computation coding first abstracts Gaussian networks into the corresponding modulo sum multiple-access channels via nested lattice codes and linear network coding and then computes the desired function by using linear Slepian-Wolf source coding. For orthogonal Gaussian networks (with no broadcast and multiple-access components), the computation capacity is characterized for a class of networks. For Gaussian networks with multiple-access components (but no broadcast), an approximate computation capacity is characterized for a class of networks.Comment: 30 pages, 12 figures, submitted to IEEE Transactions on Information Theor

A Unified Approach for Network Information Theory

In this paper, we take a unified approach for network information theory and prove a coding theorem, which can recover most of the achievability results in network information theory that are based on random coding. The final single-letter expression has a very simple form, which was made possible by many novel elements such as a unified framework that represents various network problems in a simple and unified way, a unified coding strategy that consists of a few basic ingredients but can emulate many known coding techniques if needed, and new proof techniques beyond the use of standard covering and packing lemmas. For example, in our framework, sources, channels, states and side information are treated in a unified way and various constraints such as cost and distortion constraints are unified as a single joint-typicality constraint. Our theorem can be useful in proving many new achievability results easily and in some cases gives simpler rate expressions than those obtained using conventional approaches. Furthermore, our unified coding can strictly outperform existing schemes. For example, we obtain a generalized decode-compress-amplify-and-forward bound as a simple corollary of our main theorem and show it strictly outperforms previously known coding schemes. Using our unified framework, we formally define and characterize three types of network duality based on channel input-output reversal and network flow reversal combined with packing-covering duality.Comment: 52 pages, 7 figures, submitted to IEEE Transactions on Information theory, a shorter version will appear in Proc. IEEE ISIT 201

Infinite-message Interactive Function Computation in Collocated Networks

An interactive function computation problem in a collocated network is studied in a distributed block source coding framework. With the goal of computing a desired function at the sink, the source nodes exchange messages through a sequence of error-free broadcasts. The infinite-message minimum sum-rate is viewed as a functional of the joint source pmf and is characterized as the least element in a partially ordered family of functionals having certain convex-geometric properties. This characterization leads to a family of lower bounds for the infinite-message minimum sum-rate and a simple optimality test for any achievable infinite-message sum-rate. An iterative algorithm for evaluating the infinite-message minimum sum-rate functional is proposed and is demonstrated through an example of computing the minimum function of three sources.Comment: 5 pages. 2 figures. This draft has been submitted to IEEE International Symposium on Information Theory (ISIT) 201