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.

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We initiate the study of a natural and practically relevant new variant of online caching where the to-be-cached items can have dependencies. We assume that the universe is a tree T and items are tree nodes; we require that if a node v is cached then the whole subtree T (v) rooted at v is cached as well. This theoretical problem finds an immediate application in the context of forwarding table optimization in IP routing and software-defined networks.We present an elegant online deterministic algorithm TC for this problem, and rigorously prove that its competitive ratio is O(height(T ) · k ONL /(k ONL − k OPT + 1)), where k ONL and k OPT denote the cache sizes of an online and the optimal offline algorithm, respectively. The result is optimal up to a factor of O(height(T )).

Abstract-Virtualized datacenters offer great flexibilities in terms of resource allocation. In particular, by decoupling applications from the constraints of the underlying infrastructure, virtualization supports an optimized mapping of virtual machines as well as their interconnecting network to their physical counterparts: essentially a graph embedding problem.However, existing embedding algorithms such as Oktopus and Proteus often ignore a crucial dimension of the embedding problem, namely data locality: the input to a cloud application such as MapReduce is typically stored in a distributed, and sometimes redundant, file system. Since moving data is costly, an embedding algorithm should be data locality aware, and allocate computational resources close to the data; in case of redundant storage, the algorithm should also optimize the replica selection.This paper initiates the algorithmic study of data locality aware virtual cluster embeddings on datacenter topologies. We show that despite the multiple degrees of freedom in terms of embedding, replica selection and assignment, many problems can be solved efficiently. We also highlight the limitations of such optimizations, by presenting several NP-hardness proofs; interestingly, our hardness results also hold in uncapacitated networks of small diameter.

By enhancing the traditional static network (e.g., based on electric switches) with a dynamic topology (e.g., based on reconfigurable optical switches), emerging reconfigurable data centers introduce unprecedented flexibilities in how networks can be optimized toward the workload they serve. However, such hybrid data centers are currently limited by a restrictive routing policy enforcing artificial segregation: each network flow can only use either the static or the flexible topology, but not a combination of the two. This paper explores the algorithmic problem of supporting more general routing policies, which are not limited by segregation. While the potential benefits of non-segregated routing have been demonstrated in recent work, the underlying algorithmic complexity is not well-understood. We present a range of novel results on the algorithmic complexity of non-segregated routing. In particular, we show that in certain specific scenarios, optimal data center topologies with nonsegregated routing policies can be computed in polynomial-time. In many variants of the problem, however, introducing a more flexible routing comes at a price of complexity: we prove several important variants to be NP-hard.