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.
In wireless sensor networks, one is not interested in downloading all the data from all the sensors. Rather, one is only interested in collecting from a sink node a relevant function of the sensor measurements. This paper studies the maximum rate at which functions of sensor measurements can be computed and communicated to the sink node. It focuses on symmetric functions, where only the data from a sensor is important, not its identity. The results include the following. The maximum rate of downloading the frequency histogram in a random planar multihop network with n nodes is O(1 logn) A subclass of functions, called type-sensitive functions, is maximally difficult to compute. In a collocated network, they can be computed at rate O(1 n), and in a random planar multihop network at rate O(1 logn). This class includes the mean, mode, median, etc. Another subclass of functions, called type-threshold functions, is exponentially easier to compute. In a collocated network they can be computed at rate O(1 logn), and in a random planar multihop network at rate O(1 loglogn). This class includes the max, min, range, etc. The results also show the architecture for processing information across sensor networks.  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. A sender communicates with a receiver who wishes to reliably evaluate a function of their combined data. We show that if only the sender can transmit, the number of bits required is a conditional entropy of a naturally defined graph. We also determine the number of bits needed when the communicators exchange two messages. Reference is made to the results of rate distortion in evaluating the function of two random variables.