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.
The problem of worst-case block function computation was formulated in @cite_9 . The authors identify two classes of symmetric functions namely functions exemplified by Mean and Median, and functions, exemplified by Maximum and Minimum. The maximum rates for computation of type-sensitive and type-threshold functions in random planar networks are shown to be @math and @math respectively, for a network of @math nodes. A communication complexity approach was used to establish upper bounds on the rate of computation in collocated networks.
@cite_9
2123976939
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.