Time and Energy Complexity of Distributed Computation of a Class of Functions in Wireless Sensor Networks

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, and 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 1) the communication topology assumed and 2) 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 centralized contention-free scheduling of packet transmissions. First, 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  ffiffiffiffiffiffiffi n log n

Quick, Decentralized, Energy-Efficient One-Shot Max Function Computation Using Timer-Based Selection

“…Network Types and Channel Models: Broadcast or colocated networks are considered in [3], [6]- [8] and multi-hop networks in [2]- [5], [7]- [12]. The papers also differ in the channel models assumed.…”

Maximum Achievable Throughput in a Wireless Sensor Network Using In-Network Computation for Statistical Functions

“…They show that in O( n/ log n) slots, MAX can be made available at the sink with high probability. This is interesting as the best coordinated protocol given in [16] also takes the same O( n/ log n) slots to compute MAX. Most-Aoyama and Shah [17] give an elegant distributed scheme based on exponential random variables to compute the class of separable functions such as SUM in a wireless network using gossip protocols.…”

“…Averaging out the constraints (8-14) using these definitions is straightforward. However, time also appears in the interference constraints (15)(16). It is possible in principle to average them too but the resulting flow model is very inaccurate since the interference model is averaged out.…”

“…Instead, we reformulate these constraints using the extensive formulation of the problem. First, we generate all the ISets compatible with (15)(16). This is equivalent to computing the link-Iset incidence matrix…”

“…Note that all the links in any given Iset satisfy the interference constraints by construction and hence the original interference constraints (15)(16) are redundant in the problem formulation.…”