Protocol design for dynamic Delaunay triangulation
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Delaunay triangulation (DT) is a useful geometric structure for net-working applications. We define a distributed DT and present a necessary and sufficient condition for a distributed DT to be correct. This condition is used as a guide for protocol design. We investigate the design of join, leave, failure, and maintenance protocols for a set of nodes in d-dimension (d > 1) to construct and maintain a distributed DT in a dynamic environment. The join, leave, and failure protocols in the suite are proved to be correct for a single join, leave, and failure, respectively. For a system under churn, it is impossible to maintain a correct distributed DT continually. We define an accuracy metric such that accuracy is 100% if and only if the distributed DT is correct. The suite also includes a maintenance protocol designed to recover from incorrect system states and to improve accuracy. In designing the protocols, we make use of two novel observations to substantially improve protocol efficiency. First, in the neighbor discovery process of a node, many replies to the node's queries contain redundant information. Second, the use of a new failure protocol that employs a proactive approach to recovery is better than the reactive approaches used in prior work. Experimental results show that our new suite of protocols maintains high accuracy for systems under churn and each system converges to 100% accuracy after churning stopped. They are much more efficient than protocols in prior work. To illustrate the usefulness of distributed DT for networking applications, we also present several application protocols including greedy routing, finding a closest existing node, clustering, broadcast, and geocast. Bose and Morin proved in 2004 that greedy routing always succeeds to find the destination node on a DT. We prove that greedy routing always finds a closest existing node to a given point, and our broadcast and geocast protocols always deliver a message to every target node. Our broadcast and geocast protocols are also efficient in the sense that very few target nodes receive duplicate messages, and non-target nodes receive no message. Performance characteristics of greedy routing, broadcast, and geocast are investigated using simulation experiments. We also investigate the impact of inaccurate coordinates on the performance of greedy routing, broadcast, and geocast.