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 Steiner 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.