Afek, Awerbuch, Plotkin, and Saks identified an important fundamental problem inherent to distributed networks, which they called the Resource Controller problem. Consider, first, the problem in which one node (called the 'root') is required to estimate the number of events that occurred all over the network. This counting problem can be viewed as a useful variant of the heavily studied and used task of topology update (that deals with collecting all remote information). The Resource Controller problem generalizes the counting problem: such remote events are considered as requests, and the counting node, i.e., the 'root', also issues permits for the requests. That way, the number of request granted can be controlled (bounded). An efficient Resource Controller was constructed in the paper by Afek et al., which can operate on a dynamic network assuming that the network is spanned by a tree that may only grow, and only by allowing leaves to join the tree. In contrast, the Resource Controller presented here can operate under a more general dynamic model, allowing the spanning tree of the network to undergo both insertions and deletions of both leaves and internal nodes. Despite the more dynamic network model we allow, the message complexity of our controller is always at most the message complexity of the more restricted controller. All the applications for the controller of Afek et al. apply also for our controller. Moreover, with the same message complexity, our controller can handle these applications under the more general dynamic model mentioned above. In particular, under the more general dynamic model, the new controller can be transformed into an efficient sizeestimation protocol, i.e., a protocol allowing the root to maintain a constant estimation of the number of nodes in * Supported in part by a grant from the Ali Kaufmann PostDoc fellowship and by a grant from the Israeli Ministry of Science and Technology. † Supported in part by a grant from the Israel Science Foundation. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. the dynamically changing network. Informally, the resulted new size-estimation protocol uses O(log 2 n) amortized message complexity per topological change (assuming that the number of changes in the network size is "not too small"), where n is the current number of nodes in the network. In addition, with the same message complexity (as that of the size-estimation protocol), the new controller can be used to solve the name-assignment problem by assigning and maintaining unique log n + O(1)-bit identifiers for the nodes of the dynamic network. The new size-estimation protocol can be used for other applications, not mentioned in the paper by Afek et al.. Specifically, it can be used to extend many existing labeling schemes supporting different queries (e.g routing, ancestry, etc.) so that these schemes can now operate correctly also under more general models. These extensions maintain the same asymptotic sizes of the corresponding labels (or routing tables) of the original schemes and incur only a relatively low extra additive cost to the message complexity of the corresponding original schemes.
Korman and Kutten REF introduce an (M, W )-controller with a similar message complexity 2 which supports all three types of topology changes (i.e., the insertion of leaves, the insertion of internal nodes, and the deletion of nodes).
Controller and estimator for dynamic networks
PODC '07
Computer Science