General Compact Labeling Schemes for Dynamic Trees

Abstract

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.

A general method for constructing labeling schemes for dynamic trees was previously developed in [28]. 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, routing, nearest common ancestor etc.. The resulted 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 [28], 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 on expense of communication.

Informally, for any k we present a dynamic labeling scheme incurring multiplicative overhead factors (over the static scheme) of O(log _k n) on the label size and O(klog _k n) on the amortized message complexity. In particular, by setting $k = \sqrt{n}$, we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for the routing and the nearest common ancestor functions.