Local Distributed Algorithms in Highly Dynamic Networks

We define a generalization of local distributed graph problems to (synchronous round-based) dynamic networks and present a framework for developing algorithms for these problems. The algorithms should satisfy non-trivial guarantees in every round. The guarantees should be stronger the more stable the graph has been during the last few rounds and coincide with the definition of the static graph problem if no topological change appeared recently. Moreover, if only a constant neighborhood around some part of the graph is stable during an interval, the algorithms should quickly converge to a solution for this part of the graph that remains unchanged throughout the interval. We demonstrate our generic framework with two classic distributed graph problems, namely (degree+1)-vertex coloring and maximal independent set (MIS). To illustrate the given guarantees consider the vertex coloring problem: Any conflict between two nodes caused by a newly inserted edge is resolved within T=O(log n) rounds. During this conflict resolving both nodes always output colors that are not in conflict with their respective 'old' neighbors. The largest color that a node is allowed to output is determined by the number of distinct neighbors that it has seen in the last T rounds.