Improved Compact Routing Schemes for Dynamic Trees

A classical routing problem consists of assigning a label and distinct port numbers to each node of a graph, such that for every node nu, given its own label and the label of any destination vertex u, node nu can find which of its incident port numbers leads, to the next vertex on a shortest path connecting nu and u. hi the static (fixed topology) setting, such a, routing scheme is evaluated by the label size, i.e., the maximal number of bits stored in a label. Naturally, special attention is given to compact schemes, which are schemes enjoying asymptotically optimal labels. Many routing schemes were proposed for the static setting. However, the more realistic and complex dynamic setting, in which topology changes may occur at arbitrary nodes, has received much less attention. In the dynamic setting, the occurrence of topology changes may force the scheme to occasionally update the (hopefully short) labels, by delivering information from place to place. This raises a natural tradeoff between the size of the labels and the number of messages required for maintaining them. The above dynamic routing problem was proposed by Afek, Gafni, and Ricklin (1989), who also presented an elegant and rather efficient dynamic routing scheme for trees, supporting one type of topology change, namely, the addition of a leaf. Various attempts for improving the tradeoff between the label size and the message complexity as well as for supporting more types of topology changes on trees, were subsequently proposed. Still, the best known compact routing scheme for dynamic trees has' very high message complexity, namely, O(n(epsilon)) amortized messages per topological change. Moreover, previous routing schemes for dynamic trees support at most two kinds of topology changes, namely, the addition and the removal of a leaf node. In this paper, we present two compact routing schemes for dynamic trees that incur extremely low message complexity and can support more types of topology changes that) previous schemes. We first present a dynamic compact routing scheme that supports the additions of both leaves and internal nodes and incurs only O(log n) amortized message complexity per node. We then extend that scheme obtaining a dynamic compact routing scheme that supports additions of both leaves and internal nodes as well as deletions of nodes of degree at most 2. The extended scheme incurs O(log(2) n) amortized message complexity per topological change.