Online Balanced Repartitioning

This paper initiates the study of a fundamental online problem called online balanced repartitioning. Unlike the classic graph partitioning problem, our input is an arbitrary sequence of communication requests between nodes, with patterns that may change over time. The objective is to dynamically repartition the n nodes into ` clusters, each of size k. Every communication request needs to be served either locally (cost 0), if the communicating nodes are collocated in the same cluster, or remotely (cost 1), using intercluster communication, if they are located in different clusters. The algorithm can also dynamically update the partitioning by migrating nodes between clusters at cost α per node migration. Therefore, we are interested in online algorithms which find a good tradeoff between the communication cost and the migration cost, maintaining partitions which minimize the number of inter-cluster communications. We consider settings both with and without cluster-size augmentation. For the former, we prove a lower bound which is strictly larger than k, which highlights an interesting difference to online paging. Somewhat surprisingly, and unlike online paging, we prove that any deterministic online algorithm has a non-constant competitive ratio of at least k, even with augmentation. Our main technical contribution is an O(k log k)-competitive algorithm for the setting with (constant) augmentation. We believe that our model finds interesting applications, e.g., in the context of datacenters, where virtual machines need to be dynamically embedded on a set of (multi-core) servers, and where machines migrations are possible, but costly.