优先级度量结构和嵌入
Prioritized Metric Structures and Embedding
摘要：
度量数据结构（距离预言、距离标记方案、路由方案）和低失真嵌入提供了强大的算法方法，其已成功地应用于近似算法\cite{llr}、在线算法\cite{BBMN11}、分布式算法\cite{KKMPT12}和用于计算稀疏因子\cite{ST04}。然而，这种方法似乎具有限制：最坏情况的性能固有地取决于度量的基数，并且不能预先指定哪些顶点/点应该享受更好的服务（即，拉伸/变形、标签尺寸/尺寸）。在本文中，我们减轻了这一限制，设计了一套{ \em优先}度量数据结构和嵌入。我们表明，给定图顶点（分别为度量点）的优先级排序$(x_1,x_2,\ldots,x_n)$，可以设计一个度量数据结构（分别为嵌入），其中拉伸（分别为，由包含顶点$x_j$的任何对引起的失真）将取决于顶点的秩$j$。我们还表明，其他重要的参数，如标签的大小和（在某种意义上）的尺寸，可能只取决于$j$。在我们的一些度量数据结构中（分别是，嵌入）我们实现了两个优先级的拉伸（分别，失真）和标签尺寸（分别，维度）{ \em同时}。我们的度量数据结构和嵌入的最坏情况下的性能通常是渐近不差于他们的非优先级的同行。
英文摘要：
Metric data structures (distance oracles, distance labeling schemes, routing schemes) and low-distortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms \cite{llr}, online algorithms \cite{BBMN11}, distributed algorithms \cite{KKMPT12} and for computing sparsifiers \cite{ST04}. However, this methodology appears to have a limitation: the worst-case performance inherently depends on the cardinality of the metric, and one could not specify in advance which vertices/points should enjoy a better service (i.e., stretch/distortion, label size/dimension) than that given by the worst-case guarantee. In this paper we alleviate this limitation by devising a suit of {\em prioritized} metric data structures and embeddings. We show that given a priority ranking $(x_1,x_2,\ldots,x_n)$ of the graph vertices (respectively, metric points) one can devise a metric data structure (respectively, embedding) in which the stretch (resp., distortion) incurred by any pair containing a vertex $x_j$ will depend on the rank $j$ of the vertex. We also show that other important parameters, such as the label size and (in some sense) the dimension, may depend only on $j$. In some of our metric data structures (resp., embeddings) we achieve both prioritized stretch (resp., distortion) and label size (resp., dimension) {\em simultaneously}. The worst-case performance of our metric data structures and embeddings is typically asymptotically no worse than of their non-prioritized counterparts.
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