We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements with zero error. We allow for block computation to enhance data fusion efficiency, and determine the minimum worst-case total bits to be communicated to perform the desired computation. We restrict attention to the class of symmetric Boolean functions, which only depend on the number of 1s among the n measurements. We define three classes of functions, namely threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and an order-optimal strategy with optimal preconstant for interval functions. Using these results, we can characterize the complexity of computing percentile type functions, which is of great interest. In our analysis, we use lower bounds from communication complexity theory, and provide an achievable scheme using information theoretic tools.
As argued in @cite_9 , an information-theoretic formulation of this problem combines the complexity of source coding with rate distortion as well as the manifold collaborative possibilities in wireless, together with the complications introduced by the function structure. There is little or no work that addresses this most general framework. One special case, a source coding problem for function computation with side information, has been studied in @cite_3 . Recently, the rate region for multi-round interactive function computation has been characterized for two nodes @cite_4 , and for collocated networks @cite_7 .
@cite_9 @cite_4 @cite_7 @cite_3
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We address the problem of compressing correlated distributed sources, i.e., correlated sources which are not co-located or which cannot cooperate to directly exploit their correlation. We consider the related problem of compressing a source which is correlated with another source that is available only at the decoder. This problem has been studied in the information theory literature under the name of the Slepian-Wolf (1973) source coding problem for the lossless coding case, and as "rate-distortion with side information" for the lossy coding case. We provide a constructive practical framework based on algebraic trellis codes dubbed as DIstributed Source Coding Using Syndromes (DISCUS), that can be applicable in a variety of settings. Simulation results are presented for source coding of independent and identically distributed (i.i.d.) Gaussian sources with side information available at the decoder in the form of a noisy version of the source to be coded. Our results reveal the promise of this approach: using trellis-based quantization and coset construction, the performance of the proposed approach is 2-5 dB from the Wyner-Ziv (1976) bound. We study the limits of communication efficiency for function computation in collocated networks within the framework of multi-terminal block source coding theory. With the goal of computing a desired function of sources at a sink, nodes interact with each other through a sequence of error-free, network-wide broadcasts of finite-rate messages. For any function of independent sources, we derive a computable characterization of the set of all feasible message coding rates - the rate region -in terms of single-letter information measures. We show that when computing symmetric functions of binary sources, the sink will inevitably learn certain additional information which is not demanded in computing the function. This conceptual understanding leads to new improved bounds for the minimum sum-rate. The new bounds are shown to be orderwise better than those based on cut-sets as the network scales. The scaling law of the minimum sum-rate is explored for different classes of symmetric functions and source parameters. This paper investigates the compress-and-forward scheme for an uplink cloud radio access network (C-RAN) model, where multi-antenna base stations (BSs) are connected to a cloud-computing-based central processor (CP) via capacity-limited fronthaul links. The BSs compress the received signals with Wyner-Ziv coding and send the representation bits to the CP; the CP performs the decoding of all the users’ messages. Under this setup, this paper makes progress toward the optimal structure of the fronthaul compression and CP decoding strategies for the compress-and-forward scheme in the C-RAN. On the CP decoding strategy design, this paper shows that under a sum fronthaul capacity constraint, a generalized successive decoding strategy of the quantization and user message codewords that allows arbitrary interleaved order at the CP achieves the same rate region as the optimal joint decoding. Furthermore, it is shown that a practical strategy of successively decoding the quantization codewords first, then the user messages, achieves the same maximum sum rate as joint decoding under individual fronthaul constraints. On the joint optimization of user transmission and BS quantization strategies, this paper shows that if the input distributions are assumed to be Gaussian, then under joint decoding, the optimal quantization scheme for maximizing the achievable rate region is Gaussian. Moreover, Gaussian input and Gaussian quantization with joint decoding achieve to within a constant gap of the capacity region of the Gaussian multiple-input multiple-output (MIMO) uplink C-RAN model. Finally, this paper addresses the computational aspect of optimizing uplink MIMO C-RAN by showing that under fixed Gaussian input, the sum rate maximization problem over the Gaussian quantization noise covariance matrices can be formulated as convex optimization problems, thereby facilitating its efficient solution. In this paper, we consider the problem of multifunctional compression with side information. The problem is how we can compress a source X so that the receiver is able to compute some deterministic functions f1(X, Y1), ..., fm(X, Ym), where Yi, 1 ≤ i ≤ m, are available at the receiver as side information. In [1], Wyner and Ziv considered this problem for the special case of m = 1 and f1(X, Y1) = X and derived a rate-distortion function. Yamamoto extended this result in [2] to the case of having one general function f1(X, Y1). Both of these results were in terms of an auxiliary random variable. For the case of zero distortion, in [3], Orlitsky and Roche gave an interpretation of this variable in terms of properties of the characteristic graph which led to a particular coding scheme. This result was extended in [4] by providing an achievable scheme based on colorings of the characteristic graph. In a recent work, reference [5] has considered this problem for a general tree network where intermediate nodes are allowed to perform some computations. These previous works only considered the case where the receiver only wants to compute one function (m=1). Here, we want to consider the case in which the receiver wants to compute several functions with different side information random variables and zero distortion. Our results do not depend on the fact that all functions are desired in one receiver and one can apply them to the case of having several receivers with different desired functions (i.e., functions are separable). We define a new concept named the multi-functional graph entropy which is an extension of the graph entropy defined by Korner in [6]. We show that the minimum achievable rate for this problem is equal to the conditional multi-functional graph entropy of random variable X given side informations. We also propose a coding scheme based on graph colorings to achieve this rate.