Collecting Correlated Data through a Network with Minimum Cost: Distance Entropy and a Practical Order Optimal Design

We study the communication cost of collecting correlated data at a sink over a network. This is motivated by the correlated data gathering in sensor networks and the internet. In many energy-constrained wireless sensor networks, nodes cooperatively forward correlated sensed data to data sinks. For the internet case, huge amount of correlated network traces need to be collected for management purposes. In order to reduce the communication cost (e.g. overall energy, bandwidth, etc.) used for data collection, previous works have focused on specific coding schemes, such as Slepian-Wolf Code (SWC) or Explicit Entropy Code (EEC). However, the minimum communication cost under arbitrary coding/routing schemes has not yet been characterized. In this article, we prove that the minimum communication cost can be achieved using Slepian-Wolf Code and Commodity Flow Routing when the link communication cost is a convex function of link data rate. Furthermore, we find it useful to introduce a new metric distance entropy, a generalization of entropy, to characterize the data collection limit of networked sources. When the energy consumption is proportional to the link data rate (e.g. normally in 802.11), we show that distance entropy provides a lower bound of the communication cost and can be achieved by using a specific rate SWC and shortest path routing. Theoretically, achieving optimality may require global knowledge of the data correlation structure, which may not be available in practice. Therefore, we propose a simple, hierarchical scheme that primarily exploits data correlation between local neighboring nodes. We show that for several generic correlation structures, the communication cost achieved by this scheme is within a constant factor of the distance entropy, i.e., it is order optimal. This order optimality is shown for two deployment strategies: a 2D grid regular network and a 2D Poisson process random network. Finally, we simulate our algorithm using radar reflectivity data as well as traces from Gaussian Markov Fields (GMF). As the network size goes large, for the radar data, we find our algorithm saves two thirds of the communication cost compared to a non-coding approach; as for the GMF data, our algorithm converges to a constant factor ( ) of the distance entropy.