Differential Forms for Target Tracking and Aggregate Queries in Distributed Networks

Consider mobile targets in a plane and their movements being monitored by a network such as a field of sensors. We develop distributed algorithms for in-network tracking and range queries for aggregated data (for example, returning the number of targets within any user given region). Our scheme stores the target detection information locally in the network and answers a query by examining the perimeter of the given range. The cost of updating data about mobile targets is proportional to the target displacement. The key insight is to maintain in the sensor network a function with respect to the target detection data on the graph edges that is a differential form such that the integral of this form along any closed curve C gives the integral within the region bounded by C. The differential form has great flexibility, making it appropriate for tracking mobile targets. The basic range query can be used to find a nearby target or any given identifiable target with cost O(d), where d is the distance to the target in question. Dynamic insertion, deletion, coverage holes, and mobility of sensor nodes can be handled with only local operations, making the scheme suitable for a highly dynamic network. It is extremely robust and capable of tolerating errors in sensing and target localization. Targets do not need to be identified for the tracking, thus user privacy can be preserved. In this paper, we only elaborate the advantages of differential forms in tracking of mobile targets. Similar routines can be applied for organizing many other types of information-for example, streaming scalar sensor data (such as temperature data field)-to support efficient range queries. We demonstrate through analysis and simulations that this scheme compares favorably to existing schemes that use location services for answering aggregate range queries of target detection data.