General Compact Labeling Schemes for Dynamic Trees

TL;DR

This paper introduces a general method to extend static tree labeling schemes to dynamic trees, enabling efficient computation of functions like ancestry and routing in changing networks with minimal overhead.

Contribution

It presents a universal approach to adapt static tree labeling schemes for dynamic environments, achieving near-optimal label sizes and sublinear message complexity.

Findings

Overheads are logarithmic in label size and message complexity.

Dynamic schemes are asymptotically optimal for label size.

Achieves sublinear amortized message complexity for various functions.

Abstract

Let $F$ be a function on pairs of vertices. An {\em $F$- labeling scheme} is composed of a {\em marker} algorithm for labeling the vertices of a graph with short labels, coupled with a {\em 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 {\em distributed dynamic} labeling schemes. This paper investigates labeling schemes for dynamic trees. This paper presents a general method for constructing labeling schemes for dynamic trees. Our method is based on extending an existing {\em static} tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Our resulting dynamic schemes incur overheads (over the static scheme) on the label size and on the communication complexity. 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 $k(n)=n^{\epsilon}$ for any $0<\epsilon<1$, we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for all the above mentioned functions.