Abstracts - 2007
Self-Stabilizing Hierarchical Tracking
Tina Nolte, Murat Demirbas, Nancy Lynch & Anish Arora
In this work, we addressed the problem of providing "network middleware support" for tracking a mobile object in fault-prone wireless sensor networks. We assume the availability of an underlying service for detection and classification of a mobile object [6, 11, 7]. Our tracking service solution maintains a distributed indexing structure over the sensor network and provides a basis for higher level applications such as multiple target tracking [9, 10] and pursuer-evader applications .
We designed a local and fault-tolerant tracking service for sensor networks that enables any node in the network to initiate a request to find a mobile object. In addition, we introduced a hierarchy-based concept of fault-level local stabilization as well as a scheme for achieving such stabilization.
Our tracking service is local and scalable in the sense that a find operation that is invoked within physical distance d of the mobile object requires O(d) amount of work, or communication cost, to intercept the mobile object, and moves of the mobile object to a total of at most distance d away require O(d * log(network diameter)) amount of work to update the tracking structure. There is a tradeoff between the work required for find and move operations. For example, for a 2-D network, one extreme is the evader-centric tracking program (full information strategy) where a find operation costs O(1) and a move operation of distance d costs O(d*d). The other extreme is the pursuer-centric program (no information strategy) where a move operation is O(1) and a find operation of an object d away is O(d*d). In our work we optimize both operations using a partial information strategy.
Our tracking service, defined on a hierarchy, is fault-level local stabilizing: starting from an arbitrarily corrupted state the tracking structure satisfies its specification within work proportional to the highest corrupted level of the network and number of corruptions.
Overview of the Algorithm
We achieve scalability of our tracking service by maintaining tracking information with accuracy related to the distance from the mobile object . Nearby nodes that are relatively cheap to update have more recent and accurate information about the object, whereas far away nodes that are relatively expensive to update have older and more approximate information.
Tracking structure: We assume a hierarchical partitioning service that partitions the sensor network into clusters based on radius. Our tracking structure is a path rooted at the topmost level of the hierarchy. Each process in the tracking path has at most one child, either at its level or one below it in the hierarchy, and the mobile object resides at the leaf of the tracking path. Each process in the path points to a process that is generally closer to the mobile object and has more recent information about the object.
Find operation: A find operation invoked at some process i queries processes at increasingly higher levels of the clustering hierarchy until it encounters a process on the tracking path. Once the tracking path is found, the find operation follows it to its leaf to reach the mobile object. Our find operations are local: a find invoked at a process d away from the object requires O(d) amount of work to find the object.
Move operation: We implement move-triggered updates by means of two actions, grow and shrink. The grow action enables a new path to grow from a new evader location to increasingly higher levels of the clustering hierarchy and connect to the original path. The shrink action serves to clean branches deserted by the mobile object and climbs from the lowest level to higher levels. When a mobile object moves to d distance away, 0(d * log(network diameter)) work is required to update the tracking structure.
Fault-level local stabilization: In the presence of arbitrary state corruption of processes, grow and shrink actions re-establish the tracking path in a fault-level local manner through the use of process timers. Faults are propagated at most a constant number of levels in the network.
We have completed the algorithm, described above, that tracks a mobile object in the network and have proven that, under certain hierarchy assumptions and reasonable timing assumptions, the algorithm successfully enables a client process at a location initially at most distance d from a moving evader to reach the evader in O(d) time and work. Under these same assumptions we have proven that after a mobile object has moved a total of at most d distance the algorithm performs amortized work of up to O(d * # hierarchy levels) to complete updating the tracking structure. We also proved that the algorithm has the property of fault-level local stabilization.
An extended abstract of this work was published in the 8th International Conference on Principles of Distributed Systems (OPODIS 2004). We have also developed proofs of correctness that were omitted in the extended abstract. The full version can be found at .
Recently, we have examined a variant of this algorithm, VineStalk , implemented over a programming abstraction that includes fault-tolerant emulated machines called Virtual Stationary Automata (VSAs), described in . The tracking structure is very similar to the one described here, but clusters are represented by stationary VSAs instead of individual, failure-prone and mobile clients. In addition, the clustering hierarchy used is less restrictive than that used in the work described above; in particular, grid networks are allowed. For simplicity, this new work did not consider self-stabilization; we hope to rectify this in future work.
A natural extension to this work is modifying the algorithm and tracking structure to accommodate tracking of multiple evaders by multiple pursuers. This presents a planning problem, as it is likely that the best algorithms for these situations will require coordination between pursuers. A solution to such a problem could be complicated by the occurrence of corruptions in the network.
It might be possible to loosen assumptions on evader movement that we made in this work. For example, it would be desirable to design an algorithm tolerant of evader "black-outs," times where no processes in the network detect the evader. Subject to restrictions on how long such black-outs may occur, it seems possible that the fault-tolerant algorithm here can be extended to cover such cases.
Lastly, the scheme used here for accomplishing fault level local stabilization was very useful in containing faults in the network. How general is this scheme? What other hierarchy based solutions to problems could benefit from the self-stabilization tools employed in this work? Are there other kinds of natural local stabilization definitions with general means of accomplishing them?
This research is supported in part by DARPA Award number F33615-O1-C-1896, NSF contract CCR-0121277, NSF Award number CCR-0326277, USAF,AFRL Award number FA9550-04-1-0121, AFOSR contract F49620-OO-1-0097, and NSF-Texas Engineering Experiment Station grant 64961-CS.
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 Murat Demirbas, Tina Nolte, Anish Arora, and Nancy Lynch. A Hierarchy-based Fault-local Stabilizing Algorithm for Tracking in Sensor Networks. In 8th International Conference on Principles of Distributed Systems (OPODIS 2004), December 2004.
 Murat Demirbas, Tina Nolte, Anish Arora, and Nancy Lynch. STALK: A Self-Stabilizing Hierarchical Tracking Service for Sensor Networks. Technical Report OSU-CISRC-4/03-TR19, OSU, 2003.
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