General compact labeling schemes for dynamic trees

Amos Korman
2007 Distributed computing  
Let F be a function on pairs of vertices. An F -labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F (u, v) of any two vertices u and v directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. This paper investigates labeling schemes for dynamic trees. We
more » ... sider two dynamic tree models, namely, the leaf-dynamic tree model in which at each step a leaf can be added to or removed from the tree and the leaf-increasing tree model in which the only topological event that may occur is that a leaf joins the tree. A general method for constructing labeling schemes for dynamic trees (under the above mentioned dynamic tree models) was previously developed in [29] . This method is based on extending an existing static tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as distance, separation level, ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Their resulting dynamic schemes incur overheads (over the static scheme) on the label size and on the communication complexity. In particular, all their schemes yield a multiplicative overhead factor of Ω(log n) on the label sizes of the static schemes. Following [29], we develop a different general method for extending static labeling schemes to the dynamic tree settings. Our method fits the same class of tree functions. In contrast to the above paper, our trade-off is designed to minimize the label size, sometimes at the expense of communication. Informally, for any function k(n) and any static F -labeling scheme on trees, we present an F -labeling scheme on dynamic trees incurring multiplicative overhead factors (over the static scheme) of O(log k(n) n) on the label size and O(k(n) log k(n) n) on the amortized message complexity. In particular, by setting k(n) = n for any 0 < < 1, we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for the ancestry relation, the id-based and label-based nearest common ancestor relation and the routing function. Recently, a number of studies focused on a localized network representation method based on assigning a (hopefully short) label to each vertex, allowing one to infer information about any two vertices directly from their labels, without using any additional information sources. Such labeling schemes have been developed for a variety of information types, including vertex adjacency [8, 7, 22] , distance [30, 27, 19, 18, 16, 23, 35 , 10, 2], tree routing [13, 36], flow and connectivity [26], tree ancestry [5, 6, 25], nearest common ancestor in trees [3] and various other tree functions, such as center, separation level, and Steiner weight of a given subset of vertices [31]. See [17] for a survey. By now, the basic properties of localized labeling schemes for static (fixed topology) networks are reasonably well-understood. In most realistic contexts, however, the typical setting is highly dynamic, namely, the network topology undergoes repeated changes. Therefore, for a representation scheme to be practically useful, it should be capable of reflecting online the current up-to-date picture in a dynamic setting. Moreover, the algorithm for generating and revising the labels must be distributed, in contrast with the sequential and centralized label assignment algorithms described in the above cited papers. The dynamic models investigated in this paper concern the leaf-dynamic tree model in which at each step a leaf can be added to or removed from the tree and the leaf-increasing tree model in which the only topological event that may occur is that a leaf joins the tree. We present a general method for constructing dynamic labeling schemes which is based on extending existing static tree labeling schemes to the dynamic setting. This approach fits a number of natural tree functions, such as routing , ancestry relation, nearest common ancestor relation, distance and separation level. Such an extension can be naively achieved by calculating the static labeling from scratch after each topological change. Though this method yields a good label size, it may incur a huge communication complexity. Another naive approach would be that each time a leaf u is added as a child of an existing node v, the label given to u is the label of v concatenated with F (u, v). Such a scheme incurs very little communication, however, the labels may be huge. Before stating the results included in this paper, we list some previous related works.
doi:10.1007/s00446-007-0035-z fatcat:wvhz255tp5hsdadbkuna2wlw2m