A Relation Between Network Computation and Functional Index Coding Problems

In contrast to the network coding problem wherein the sinks in a network demand subsets of the source messages, in a network computation problem the sinks demand functions of the source messages. Similarly, in the functional index coding problem, the side information and demands of the clients include disjoint sets of functions of the information messages held by the transmitter instead of disjoint subsets of the messages, as is the case in the conventional index coding problem. It is known that any network coding problem can be transformed into an index coding problem and vice versa. In this work, we establish a similar relationship between network computation problems and a class of functional index coding problems, viz., those in which only the demands of the clients include functions of messages. We show that any network computation problem can be converted into a functional index coding problem wherein some clients demand functions of messages and vice versa. We prove that a solution for a network computation problem exists if and only if a functional index code (of a specific length determined by the network computation problem) for a suitably constructed functional index coding problem exists. And, that a functional index coding problem admits a solution of a specified length if and only if a suitably constructed network computation problem admits a solution.Comment: 3 figures, 7 tables and 9 page

Optimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms

We consider optimal distributed computation of a given function of distributed data. The input (data) nodes and the sink node that receives the function form a connected network that is described by an undirected weighted network graph. The algorithm to compute the given function is described by a weighted directed acyclic graph and is called the computation graph. An embedding defines the computation communication sequence that obtains the function at the sink. Two kinds of optimal embeddings are sought, the embedding that---(1)~minimizes delay in obtaining function at sink, and (2)~minimizes cost of one instance of computation of function. This abstraction is motivated by three applications---in-network computation over sensor networks, operator placement in distributed databases, and module placement in distributed computing. We first show that obtaining minimum-delay and minimum-cost embeddings are both NP-complete problems and that cost minimization is actually MAX SNP-hard. Next, we consider specific forms of the computation graph for which polynomial time solutions are possible. When the computation graph is a tree, a polynomial time algorithm to obtain the minimum delay embedding is described. Next, for the case when the function is described by a layered graph we describe an algorithm that obtains the minimum cost embedding in polynomial time. This algorithm can also be used to obtain an approximation for delay minimization. We then consider bounded treewidth computation graphs and give an algorithm to obtain the minimum cost embedding in polynomial time

A Distributed Computationally Aware Quantizer Design via Hyper Binning

We design a distributed function aware quantization scheme for distributed functional compression. We consider $2$ correlated sources $X_1$ and $X_2$ and a destination that seeks the outcome of a continuous function $f(X_1,\,X_2)$. We develop a compression scheme called hyper binning in order to quantize $f$ via minimizing entropy of joint source partitioning. Hyper binning is a natural generalization of Cover's random code construction for the asymptotically optimal Slepian-Wolf encoding scheme that makes use of orthogonal binning. The key idea behind this approach is to use linear discriminant analysis in order to characterize different source feature combinations. This scheme captures the correlation between the sources and function's structure as a means of dimensionality reduction. We investigate the performance of hyper binning for different source distributions, and identify which classes of sources entail more partitioning to achieve better function approximation. Our approach brings an information theory perspective to the traditional vector quantization technique from signal processing

Network Flows for Function Computation

We consider in-network computation of an arbitrary function over an arbitrary communication network. A network with capacity constraints on the links is given. Some nodes in the network generate data, e. g., like sensor nodes in a sensor network. An arbitrary function of this distributed data is to be obtained at a terminal node. The structure of the function is described by a given computation schema, which in turn is represented by a directed tree. We design computing and communicating schemes to obtain the function at the terminal at the maximum rate. For this, we formulate linear programs to determine network flows that maximize the computation rate. We then develop a fast combinatorial primal-dual algorithm to obtain near-optimal solutions to these linear programs. As a subroutine for this, we develop an algorithm for finding the minimum cost embedding of a tree in a network with any given set of link costs. We then briefly describe extensions of our techniques to the cases of multiple terminals wanting different functions, multiple computation schemas for a function, computation with a given desired precision, and to networks with energy constraints at nodes