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
2158565632 2123976939 2507428467 2071020383
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. 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 design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: @math Offline Hamming Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional Hamming space for @math , we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time @math or deterministic time @math . These also lead to faster MAX-SAT algorithms for sparse CNFs. @math Offline Approximate Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional @math or Euclidean space, we can find a @math -approximate nearest (or furthest) blue neighbor for each red point in randomized time near @math . @math SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for @math circuits with a subquadratic number of linear threshold gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic @math time. This also implies new circuit lower bounds for threshold circuits. We also give a randomized @math -time SAT algorithm for subexponential-size @math circuits, where the top @math gate and middle @math gates have @math fan-in. We consider the following norm ∥ƒ for Boolean functions ƒ: 0, 1 n × 0, 1 n → 0, 1 : ∥ƒ|| = max ¦M(ƒ∗)v¦, v ϵ R2n, ¦v¦ = 1 . Here, M(ƒ∗) denotes the 2n × 2n-matrix obtained by setting M(ƒ∗)x,y=1 if ƒ(x,y)=1−1 if ƒ(x,y)=0. Further, we refer to the usual product of real matrices and vectors and denote by ¦v¦ the Euclidean norm of real vectors v. In this paper it will be shown by geometric arguments that for each function ƒ: 0,1 n × 0,1 n → 0,1 the following is true. - The length of each probabilistic communication protocol which computes ƒ with error bounded by 12 − 1s, s ϵ N, can be estimated from below by 14(n − log2∥ƒ||−log2 √s−2). - The number of edges of any threshold circuit of depth two computing ƒ cannot be smaller than 2n−1‖ƒ‖ These results yield better lower bounds on probabilistic communication complexity as well as on the complexity of threshold circuits of depth two. Further, characterizing ∥ƒ∥ by the eigenvalues of M(ƒ∗) we obtain a method to construct iteratively functions which are hard to compute in the above models. This implies lower bounds for decision problems that previous probabilistic techniques from Yao (1983), Halstenberg and Reischuk (1988) and (1987) cannot be applied to.