小深度树的紧致代数标号方案
Compact Ancestry Labeling Schemes for Trees of Small Depth
摘要：
{ \em祖先标记方案}以这样的方式标记任何树的节点，即树中任何两个节点之间的祖先查询可以仅通过查看它们对应的标签来回答。评估祖先标记方案的质量的常用度量是通过其{ \em标签大小}，即在所有$n$节点树上存储在标签中的最大位数。祖先标记方案的设计在XML搜索引擎中得到应用。在这些应用中，即使是标签大小的微小改进也很重要。事实上，关于这个主题的文献感兴趣的是确切的标签大小，而不仅仅是它的数量级。因此，在提出了$2\log n$比特大小的原始方案之后，投入了大量的工作来改进标签大小的界限。现有技术的上限是$\log n + O(\sqrt{\log n})$位，其仍然远离已知的$\log n + Ω(\log\log n)$下限。此外，在添加剂$O(\sqrt{\log n})$长期隐藏的常数因子是大的，这使得该长期占主导地位的标签大小为典型的当前XML树。为了给真正的XML数据提供良好的性能，我们依赖于这样一种观察：典型XML树的深度从上到下是由一个小常数限定的。考虑到这一点，我们提出了一个祖先标记计划的大小$\log n+2\log d +O(1)$，最多$n$节点和深度最多$d$的树的家庭。除了我们的主要结果，我们证明了一个结果，可能是独立的利益有关的存在性的线性{ \em通用图}的家庭森林有界深度的树。
英文摘要：
An {\em ancestry labeling scheme} labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry labeling scheme is by its {\em label size}, that is the maximal number of bits stored in a label, taken over all $n$-node trees. The design of ancestry labeling schemes finds applications in XML search engines. In the context of these applications, even small improvements in the label size are important. In fact, the literature about this topic is interested in the exact label size rather than just its order of magnitude. As a result, following the proposal of an original scheme of size $2\log n$ bits, a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is $\log n + O(\sqrt{\log n})$ bits which is still far from the known $\log n + Ω(\log\log n)$ lower bound. Moreover, the hidden constant factor in the additive $O(\sqrt{\log n})$ term is large, which makes this term dominate the label size for typical current XML trees. In attempt to provide good performances for real XML data, we rely on the observation that the depth of a typical XML tree is bounded from above by a small constant. Having this in mind, we present an ancestry labeling scheme of size $\log n+2\log d +O(1)$, for the family of trees with at most $n$ nodes and depth at most $d$. In addition to our main result, we prove a result that may be of independent interest concerning the existence of a linear {\em universal graph} for the family of forests with trees of bounded depth.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
