Communication-optimal maintenance of replicated information

It is shown that keeping track of history allows significant improvements in the realistic model of communication complexity of dynamic network protocols. The communication complexity for solving an arbitrary graph problem is improved from Θ(E) to Θ( V), thus achieving the lower bound. Moreover, O(V) is also the amortized complexity of solving an arbitrary function (not only graph functions) defined on the local inputs of the nodes. As a corollary, it is found that amortized communication complexity, i.e. incremental cost of adapting to a single topology change, can be smaller than the communication complexity of solving the problem from scratch. The first stage in the solution is a communication-optimal maintenance of a spanning tree in a dynamic network. The second stage is the optimal maintenance of replicas of databases. An important example of this task is the problem of updating the description of the network´s topology at every node. For this problem the message complexity is improved from O(EV) to Θ(V). The improvement for a general database is even larger if the size of the database is larger than E