Maintaining Approximate Minimum Steiner Tree and k-center for Mobile Agents in a Sensor Network

We study the problem of maintaining group communication between m mobile agents, tracked and helped by n static networked sensors. We develop algorithms to maintain a O(lg n)-approximation to the minimum Sterner tree of the mobile agents such that the maintenance message cost is on average O(lg n) per each hop an agent moves. The key idea is to extract a ´hierarchical well-separated tree (HST)´ on the sensor nodes such that the tree distance approximates the sensor network hop distance by a factor of O(lg n). We then prove that maintaining the subtree of the mobile agents on the HST uses logarithmic messages per hop movement. With the HST we can also maintain O(lg n) approximate k-center for the mobile agents with the same message cost. Both the minimum Steiner tree and the k-center problems are NP-hard and our algorithms are the first efficient algorithms for maintaining approximate solutions in a distributed setting.