动态树的一般紧标号方案
General Compact Labeling Schemes for Dynamic Trees
摘要：
设$F$是顶点对上的函数。{ \em $F$ -标记方案}由用于用短标签标记图的顶点的{ \em标记}算法和允许直接从它们的标签计算任意两个顶点$u$和$v$的$F(u,v)$的{ \em解码器}算法组成。由于标记方案的应用主要涉及大型和动态变化的网络，因此研究{ \em分布式动态}标记方案是有意义的。本文研究了动态树的标号方案。本文提出了一种构造动态树标号方案的一般方法。我们的方法是基于扩展现有的{ \em静态}树标记计划的动态设置。这种方法适用于树的许多自然功能，如祖先关系，路由（在对手和设计者端口模型），最近的共同祖先等。我们所得到的动态计划招致开销（静态计划）的标签大小和通信的复杂性。非正式地，对于任何函数$k(n)$和树上的任何静态$F$ -标记方案，我们提出了动态树上的$F$ -标记方案，导致$O(\log_{k(n)} n)$在标签大小上和$O(k(n)\log_{k(n)} n)$在摊销消息复杂度上的乘法开销因子（相对于静态方案）。特别地，通过为任何$0<ε<1$设置$k(n)=n^ε$，我们获得了动态标记方案，渐近最优的标签大小和次线性摊销的消息复杂度的所有上述功能。
英文摘要：
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 $k(n)=n^ε$ for any $0<ε<1$, we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for all the above mentioned functions.
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