度量空间的多重嵌入
Multi-Embedding of Metric Spaces
摘要：
度量嵌入已成为算法设计中的常用技术。它的适用性通常取决于嵌入失真的程度。例如，将有限度量空间嵌入到树中可能需要作为其大小的函数的线性失真。使用概率度量嵌入，上界的失真减少到对数的大小。我们朝着绕过度量大小的失真下限的方向迈出了一步。我们定义了度量空间的“多重嵌入”，其中一个点映射到一组点上，同时保持多项式大小的目标度量并保持路径的失真。这样的多嵌入到超度量中获得的失真最多为O（log Delta loglog Delta），其中Delta是度量的纵横比。特别是，对于扩展图，我们能够获得恒定的失真嵌入到树中的欧米茄（log n）的下界相比，所有以前的概念嵌入。我们展示了两个优化问题的新嵌入的算法应用：组斯坦纳树和度量任务系统。
英文摘要：
Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Omega(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems.
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。
