Metric Embedding via Shortest Path Decompositions

We study the problem of embedding shortest-path metrics weighted graphs into $\ell_p$ spaces. introduce a new technique based on low-depth decompositions graph via shortest paths. The notion Shortest Path Decomposition depth is inductively defined: A (weighed) path has decomposition (SPD) $1$. General an SPD $k$ if it contains whose deletion leads to graph, each components at most $k-1$. In this paper we give $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$-distortion for $k$. This result asymptotically tight any fixed $p>1$, while $p=1$ up second order terms. As corollary result, show that having pathwidth embed with distortion $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$. For $p=1$, improves over best previous bound Lee and Sidiropoulos was exponential in $k$; moreover, other values $p$ gives first embeddings independent size $n$. Furthermore, use fact planar have $O(\log n)$ proof embeds $\ell_1$ $O(\sqrt{\log n})$. Our approach also results bounded treewidth, excluding minor.