Distributed Function Computation Research Articles

Private Remote Sources for Secure Multi-Function Computation

We consider a distributed function computation problem in which parties observing noisy versions of a remote source facilitate the computation of a function of their observations at a fusion center through public communication. The distributed function computation is subject to constraints, including not only reliability and storage but also secrecy and privacy. Specifically, 1) the function computed should remain <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">secret</i> from an eavesdropper observing the public communication and correlated observations, measured in terms of the information leaked about the arguments of the function, to ensure secrecy regardless of the exact function used; 2) the remote source should remain <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">private</i> from the eavesdropper and the fusion center, measured in terms of the information leaked about the remote source itself. We derive the exact rate regions for lossless and lossy single-function computation and illustrate the lossy single-function computation rate region for an information bottleneck example, in which the optimal auxiliary random variables are characterized for binary-input symmetric-output channels. We extend the approach to lossless and lossy asynchronous multiple-function computations with joint secrecy and privacy constraints, in which case inner and outer bounds for the rate regions that differ only in the Markov chain conditions imposed are characterized.

Precoder Design for Communication-Efficient Distributed MIMO Receivers With Controlled Peak-Average Power Ratio

We consider the problem of communicating over a relay-assisted multiple-input multiple-output (MIMO) channel with additive noise, in which physically separated relays forward quantized information to a central decoder where the transmitted message is to be decoded. We assume that channel state information is available in the transmitter and show that the design of a rational-forcing precoder - a precoder which is matched to the quantizers used in the relays - is beneficial for reducing the symbol error probability. It turns out that for such rational-forcing precoder based systems, there is natural tradeoff between the peak to average power ratio in the transmitter and the rate of communication between the relays and the central decoder. The precoder design problem is formulated mathematically, and several algorithms are developed for realizing this tradeoff. Optimality of the decoder communication rate is shown based on a result in distributed function computation. Numerical and simulation results show that a useful tradeoff can be obtained between the excess decoder communication rate and the peak-average power ratio in the transmitter.

A Classification of Functions in Multiterminal Distributed Computing

In the distributed function computation problem, dichotomy theorems, initiated by Han-Kobayashi, seek to classify functions by whether the rate regions for function computation improve on the Slepian-Wolf regions or not. In this paper, we develop a general approach to derive converse bounds on the distributed function computation problem. By using this approach, we recover the sufficiency part, i.e. the conditions such that the Slepian-Wolf regions become optimal, of the known dichotomy theorems in the two-terminal distributed computing. Furthermore, we derive an improved sufficient condition on the dichotomy theorem in the multiterminal distributed computing for the class of i.i.d. sources with the positivity condition. Finally, we derive the matching sufficient and necessary condition on the dichotomy theorem in the multiterminal distributed computing for the class of smooth sources.

Extremal Problem with Network-Diameter and -Minimum-Degree for Distributed Function Computation

Distributed function computation has a wide spectrum of major applications in distributed systems. Distributed computation over a network-system proceeds in a sequence of time-steps in which vertices update and/or exchange their values based on the underlying algorithm constrained by the time-(in)variant network-topology. Distributed computing network-systems are modeled as directed/undirected graphs with vertices representing compute elements and adjacency-edges capturing their uni- or bi-directional communication. To quantify an intuitive tradeoff between two graph-parameters: minimum vertex-degree and diameter of the underlying graph, we formulate an extremal problem with the two parameters: for all positive integers n and d, the extremal value $$\nabla (n, d)$$ denotes the least minimum vertex-degree among all connected order-n graphs with diameters of at most d. We prove matching upper and lower bounds on the extremal values of $$\nabla (n, d)$$ for various combinations of n- and d-values.

Lower-Bound Study for Function Computation in Distributed Networks via Vertex-Eccentricity

Distributed computing network-systems are modeled as graphs with vertices representing compute elements and adjacency-edges capturing their uni- or bi-directional communication. Distributed function computation has a wide spectrum of major applications in distributed systems. Distributed computation over a network-system proceeds in a sequence of time-steps in which vertices update and/or exchange their values based on the underlying algorithm constrained by the time-(in)variant network-topology. For finite convergence of distributed information dissemination and function computation in the model, we present a lower bound on the number of time-steps for vertices to receive (initial) vertex-values of all vertices regardless of underlying protocol or algorithmics in time-invariant networks via the notion of vertex-eccentricity in a graph-theoretic framework. We also address lower bounds on vertex-eccentricity and its maximum version in terms of common graph-parameters such as maximum degree, and order and size.

On Distributed Computing for Functions With Certain Structures

The problem of distributed function computation is studied, where functions to be computed is not necessarily symbol-wise. A new method to derive a converse bound for distributed computing is proposed; from the structure of functions to be computed, information that is inevitably conveyed to the decoder is identified, and the bound is derived in terms of the optimal rate needed to send that information. The class of informative functions is introduced, and, for the class of smooth sources, the optimal rate for computing those functions is characterized. Furthermore, for i.i.d. sources with joint distribution that may not be full support, functions that are composition of symbol-wise function and the type of a sequence are considered, and the optimal rate for computing those functions is characterized in terms of the hypergraph entropy. As a byproduct, our method also provides a conceptually simple proof of the known fact that computing a Boolean function may require as large rate as reproducing the entire source.

Information-Theoretic Lower Bounds for Distributed Function Computation

We derive information-theoretic converses (i.e., lower bounds) for the minimum time required by any algorithm for distributed function computation over a network of point-to-point channels with finite capacity, where each node of the network initially has a random observation and aims to compute a common function of all observations to a given accuracy with a given confidence by exchanging messages with its neighbors. We obtain the lower bounds on computation time by examining the conditional mutual information between the actual function value and its estimate at an arbitrary node, given the observations in an arbitrary subset of nodes containing that node. The main contributions include the following. First, a lower bound on the conditional mutual information via so-called small ball probabilities, which captures the dependence of the computation time on the joint distribution of the observations at the nodes, the structure of the function, and the accuracy requirement. For linear functions, the small ball probability can be expressed by Levy concentration functions of sums of independent random variables, for which tight estimates are available that lead to strict improvements over existing lower bounds on computation time. Second, an upper bound on the conditional mutual information via strong data processing inequalities, which complements and strengthens existing cutset-capacity upper bounds. Finally, a multi-cutset analysis that quantifies the loss (dissipation) of the information needed for computation as it flows across a succession of cutsets in the network. This analysis is based on reducing a general network to a line network with bidirectional links and self-links, and the results highlight the dependence of the computation time on the diameter of the network, a fundamental parameter that is missing from most of the existing lower bounds on computation time.

Graph Codes for Distributed Instant Message Collection in an Arbitrary Noisy Broadcast Network

We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide (i) fundamental limits on the required number of broadcasts of data gathering, and (ii) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, \textcolor{black}{arbitrary geometric networks and random Erd$\ddot{o}$s-R$\acute{e}$nyi networks}, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight \textcolor{black}{in order sense}. Our main techniques are three distributed encoding techniques, called graph codes, which are designed respectively for the above-mentioned three scenarios. Our work thus extends and unifies previous works such as those of Gallager [1] and Karamchandani~\emph{et. al.} [2] on number of broadcasts for distributed function computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.

Distributed Function Computation Over a Rooted Directed Tree

This paper establishes the capacity region for a class of source coding function computation setups, where sources of information are available at the nodes of a tree and where a function of these sources must be computed at its root. The capacity region holds for any function as long as the sources’ joint distribution satisfies a certain Markov criterion. This criterion is met, in particular, when the sources are independent. This result recovers the capacity regions of several function computation setups. These include the point-to-point communication setting with arbitrary sources, the noiseless multiple access network with conditionally independent sources, and the cascade network with Markovian sources.

Token-Based Function Computation with Memory

In distributed function computation, each node has an initial value and the goal is to compute a function of these values in a distributed manner. In this paper, we propose a novel token-based approach to compute a wide class of target functions to which we refer as “token-based function computation with memory” (TCM) algorithm. In this approach, node values are attached to tokens and travel across the network. Each pair of travelling tokens would coalesce when they meet, forming a token with a new value as a function of the original token values. In contrast to the coalescing random walk (CRW) algorithm, where token movement is governed by random walk, meeting of tokens in our scheme is accelerated by adopting a novel chasing mechanism. We proved that, compared to the CRW algorithm, the TCM algorithm results in a reduction of time complexity by a factor of at least $\sqrt{n/\log (n)}$ in Erdos-Renyi and complete graphs, and by a factor of $\log (n)/\log (\log (n))$ in torus networks. Simulation results show that there is at least a constant factor improvement in the message complexity of TCM algorithm in all considered topologies. Robustness of the CRW and TCM algorithms in the presence of node failure is analyzed. We show that their robustness can be improved by running multiple instances of the algorithms in parallel.

A Dichotomy of Functions in Distributed Coding: An Information Spectral Approach

The problem of distributed data compression for function computation is considered, where: 1) the function to be computed is not necessarily symbolwise function and 2) the information source has memory and may not be stationary nor ergodic. We introduce the class of smooth sources and give a sufficient condition on functions so that the achievable rate region for computing coincides with the Slepian-Wolf region (i.e., the rate region for reproducing the entire source) for any smooth sources. Moreover, for symbolwise functions, the necessary and sufficient condition for the coincidence is established. Our result for the full side-information case is a generalization of the result by Ahlswede and Csiszár to sources with memory; our dichotomy theorem is different from Han and Kobayashi's dichotomy theorem, which reveals an effect of memory in distributed function computation. All results are given not only for fixed-length coding but also for variable-length coding in a unified manner. Furthermore, for the full side-information case, the error probability in the moderate deviation regime is also investigated.

Efficient In-Network Computing with Noisy Wireless Channels

In this paper, we study distributed function computation in a noisy multihop wireless network. We adopt the adversarial noise model, for which independent binary symmetric channels are assumed for any point-to-point transmissions, with (not necessarily identical) crossover probabilities bounded above by some constant e. Each node takes an m-bit integer per instance, and the computation is activated after each node collects N readings. The goal is to compute a global function with a certain fault tolerance in this distributed setting; we mainly deal with divisible functions, which essentially cover the main body of interest for wireless applications. We focus on protocol designs that are efficient in terms of communication complexity. We first devise a general protocol for evaluating any divisible functions, addressing both one-shot (N = O(1)) and block computation, and both constant and large m scenarios. We also analyze the bottleneck of this general protocol in different scenarios, which provides insights into designing more efficient protocols for specific functions. In particular, we endeavor to improve the design for two exemplary cases: the identity function, and size-restricted type-threshold functions, both focusing on the constant m and N scenario. We explicitly consider clustering, rather than hypothetical tessellation, in our protocol design.

Worst-case asymmetric distributed function computation

We consider a distributed function computation problem where an information sink communicates with N correlated information sources to compute a given deterministic function of source data. A data-vector is drawn from a discrete and finite probability distribution and component x i is revealed to ith, , source. We address this problem in asymmetric communication scenarios where only the sink knows the distribution P. We are interested in computing the minimum number of source bits required, in the worst-case, to solve this problem. We propose the notion of functional ambiguity to carry out the worst-case information-theoretic analysis of distributed function computation problems. We establish its various characteristics and prove that it leads to a valid information measure. Then, we provide a constructive solution for the distributed function computation problem in terms of an interactive communication protocol and prove its optimality. Finally, we establish two equivalence classes of compressible and incompressible functions to classify the set of all computable multivariate functions based on the minimum number of source bits needed in the worst-case to compute a function in the distributed function computation set-up.

Linear Coding Schemes for the Distributed Computation of Subspaces

Let $X_1, ..., X_m$ be a set of $m$ statistically dependent sources over the common alphabet $\mathbb{F}_q$, that are linearly independent when considered as functions over the sample space. We consider a distributed function computation setting in which the receiver is interested in the lossless computation of the elements of an $s$-dimensional subspace $W$ spanned by the elements of the row vector $[X_1, \ldots, X_m]\Gamma$ in which the $(m \times s)$ matrix $\Gamma$ has rank $s$. A sequence of three increasingly refined approaches is presented, all based on linear encoders. The first approach uses a common matrix to encode all the sources and a Korner-Marton like receiver to directly compute $W$. The second improves upon the first by showing that it is often more efficient to compute a carefully chosen superspace $U$ of $W$. The superspace is identified by showing that the joint distribution of the $\{X_i\}$ induces a unique decomposition of the set of all linear combinations of the $\{X_i\}$, into a chain of subspaces identified by a normalized measure of entropy. This subspace chain also suggests a third approach, one that employs nested codes. For any joint distribution of the $\{X_i\}$ and any $W$, the sum-rate of the nested code approach is no larger than that under the Slepian-Wolf (SW) approach. Under the SW approach, $W$ is computed by first recovering each of the $\{X_i\}$. For a large class of joint distributions and subspaces $W$, the nested code approach is shown to improve upon SW. Additionally, a class of source distributions and subspaces are identified, for which the nested-code approach is sum-rate optimal.

Distributed Function Computation with Confidentiality

A set of terminals observe correlated data and seek to compute functions of the data using interactive public communication. At the same time, it is required that the value of a private function of the data remains concealed from an eavesdropper observing this communication. In general, the private function and the functions computed by the nodes can be all different. We show that a class of functions are securely computable if and only if the conditional entropy of data given the value of private function is greater than the least rate of interactive communication required for a related multiterminal source-coding task. A single-letter formula is provided for this rate in special cases.

Distributed Anonymous Discrete Function Computation

We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes. In this model, each node has bounded computation and storage capabilities that do not grow with the network size. Furthermore, each node only knows its neighbors, not the entire graph. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we provide a necessary condition for computability which we show to be nearly sufficient, in the sense that every function that violates this condition can at least be approximated. The problem of computing (suitably rounded) averages in a distributed manner plays a central role in our development; we provide an algorithm that solves it in time that grows quadratically with the size of the network.

Time and Energy Complexity of Distributed Computation of a Class of Functions in Wireless Sensor Networks

We consider a scenario in which a wireless sensor network is formed by randomly deploying n sensors to measure some spatial function over a field, with the objective of computing a function of the measurements and communicating it to an operator station. We restrict ourselves to the class of type-threshold functions (as defined in the work of Giridhar and Kumar, 2005), of which max, min, and indicator functions are important examples: our discussions are couched in terms of the max function. We view the problem as one of message-passing distributed computation over a geometric random graph. The network is assumed to be synchronous, and the sensors synchronously measure values and then collaborate to compute and deliver the function computed with these values to the operator station. Computation algorithms differ in (1) the communication topology assumed and (2) the messages that the nodes need to exchange in order to carry out the computation. The focus of our paper is to establish (in probability) scaling laws for the time and energy complexity of the distributed function computation over random wireless networks, under the assumption of centralized contention-free scheduling of packet transmissions. First, without any constraint on the computation algorithm, we establish scaling laws for the computation time and energy expenditure for one-time maximum computation. We show that for an optimal algorithm, the computation time and energy expenditure scale, respectively, as Theta(radicn/log n) and Theta(n) asymptotically as the number of sensors n rarr infin. Second, we analyze the performance of three specific computation algorithms that may be used in specific practical situations, namely, the tree algorithm, multihop transmission, and the Ripple algorithm (a type of gossip algorithm), and obtain scaling laws for the computation time and energy expenditure as n rarr infin. In particular, we show that the computation time for these algorithms scales as Theta(radicn/log n), Theta(n), and Theta(radicn log n), respectively, whereas the energy expended scales as , Theta(n), Theta(radicn/log n), and Theta(radicn log n), respectively. Finally, simulation results are provided to show that our analysis indeed captures the correct scaling. The simulations also yield estimates of the constant multipliers in the scaling laws. Our analyses throughout assume a centralized optimal scheduler, and hence, our results can be viewed as providing bounds for the performance with practical distributed schedulers.