计算双Lipschitz离群点嵌入直线
Computing Bi-Lipschitz Outlier Embeddings into the Line
摘要：
计算一个图形度量的双Lipschitz嵌入到最小失真的直线中的问题已经受到了很多关注。最著名的近似算法计算具有失真$O(c^2)$的嵌入，其中$c$表示最佳失真[Bethodoiu \etal ~2005]。我们提出了一个双准则近似算法，将上述结果扩展到\emph{outliers}的设置。具体地说，我们说，一个度量空间$(X,ρ)$承认一个$(k,c)$ -嵌入，如果存在$K\subset X$，与$|K|=k$，使得$(X\setminus K, ρ)$承认一个嵌入到线与失真最多$c$。给定$k\geq 0$和一个允许$(k,c)$嵌入的度量空间，对于某些$c\geq 1$，我们的算法在多项式时间内计算出一个$({\mathsf p}{\mathsf o}{\mathsf l}{\mathsf y}(k, c, \log n), {\mathsf p}{\mathsf o}{\mathsf l}{\mathsf y}(c))$嵌入。这是第一个孤立双Lipschitz嵌入算法的结果。在我们的工作之前，可比的离群值嵌入只在加性失真的情况下才知道。
英文摘要：
The problem of computing a bi-Lipschitz embedding of a graphical metric into the line with minimum distortion has received a lot of attention. The best-known approximation algorithm computes an embedding with distortion $O(c^2)$, where $c$ denotes the optimal distortion [Bădoiu \etal~2005]. We present a bi-criteria approximation algorithm that extends the above results to the setting of \emph{outliers}. Specifically, we say that a metric space $(X,ρ)$ admits a $(k,c)$-embedding if there exists $K\subset X$, with $|K|=k$, such that $(X\setminus K, ρ)$ admits an embedding into the line with distortion at most $c$. Given $k\geq 0$, and a metric space that admits a $(k,c)$-embedding, for some $c\geq 1$, our algorithm computes a $({\mathsf p}{\mathsf o}{\mathsf l}{\mathsf y}(k, c, \log n), {\mathsf p}{\mathsf o}{\mathsf l}{\mathsf y}(c))$-embedding in polynomial time. This is the first algorithmic result for outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier embeddings where known only for the case of additive distortion.
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。
