Computing Bi-Lipschitz Outlier Embeddings into the Line

Abstract

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 $(X,\rho )$ admits a $(k,c)$-embedding if there exists $K\subset X$, with $|K|=k$, such that $(X\setminus K,\rho )$ admits an embedding into the line with distortion at most $c$. Given $k\ge 0$, and a metric space that admits a $(k,c)$-embedding, for some $c\ge 1$, 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.