Maintenance of a spanning tree in dynamic networks

Many crucial network tasks such as database maintenance can be efficiently carried out given a tree that spans the network. By maintaining such a spanning tree, rather than constructing it "from-scratch" due to every topology change, one can improve the efficiency of the tree construction, as well as the efficiency of the protocols that use the tree. We present a protocol for this task which has communication complexity that is linear in the "actual" size of the biggest connected component. The time complexity of our protocol has only a polylogarithmic overhead in the "actual" size of the biggest connected component. The communication complexity of the previous solution, which was considered communication optimal, was linear in the network size, that is, unbounded as a function of the "actual" size of the biggest connected component. The overhead in the time measure of the previous solution was polynomial in the network size. In an asynchronous network it may not be clear what is the meaning of the "actual" size of the connected component at a given time. To capture this notion we define the virtual component and show that in asynchronous networks, in a sense, the notion of the virtual component is the closest one can get to the notion of the "actual" component.