Time and energy complexity of distributed computation in wireless sensor networks

Abstract

Abstract-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 [2]), 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; 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 (i) the communication topology assumed, and (ii) 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 centralised contention-free scheduling of packet transmissions. Firstly, 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 Θ n log n and Θ(n) asymptotically as the number of sensors n → ∞. Secondly, we analyze the performance of three specific computation algorithms that may be used in specific practical situations, namely, the Tree algorithm, Multi-Hop transmission, and the Ripple algorithm (a type of gossip algorithm), and obtain scaling laws for the computation time and energy expenditure as n → ∞. In particular we show that the computation time for these algorithms scales as Θ √ n log n , Θ(n) and Θ √ n log n , respectively; whereas the energy expended scales as Θ(n), Θ n n log n and Θ n √ n 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.

Topological properties of random wireless networks

“…Giridhar and Kumar [11] studies the maximum rate at which functions of sensor measurements can be computed and communicated to the sink node. Khude et al [21] study the time and energy complexity of performing distributed functions over random geometric graphs. They also consider pipelining of the functions.…”

“…A performance measure of interest is the rate at which this function can be computed over an underlying random network subject to constraints like, for example, energy consumption. In this case we could be interested in the time and energy complexity in the execution distributed function [21,11]. A canonical function is the localisation where the position of a target, a source of interest or even the nodes themselves needs to be obtained.…”

On Distributed Computation in Noisy Random Planar Networks

“…Two types of wireless networks are considered in the literature studying function computation random planar networks [1]- [3] and collocated or broadcast networks [2], [4]- [6]. Results for collocated networks are useful in solving corresponding problems for random planar networks.…”

“…Results for collocated networks are useful in solving corresponding problems for random planar networks. While [2], [3] consider noise-free links, [1], [4]- [6] consider noisy links. Computation of the histogram, and hence that of any symmetric function, in a random planar network with noisy links is considered in [1].…”

“…We ignore processing energy. In general, Et depends on rn, and hence on n. However, results in [3] indicate that for sufficiently large n (> 5000), it is reasonable to assume that the transmission energy is constant and independent of n. We follow this assumption. Note that this is different from the energy model of [1].…”