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z^!}aYWn8zh!Y`1J-H27cAu$ODVH{50NBpcFY)v|J<322U<8Jy40>yV3XSrk0$ such that for all $x,y\in X$, -\[r\leq \frac{\E[d_i(x,y)]}{d(x,y)}\leq Dr.\] +Let $(X,d)$ be a finite metric space and let $\mathcal Y=\{ (Y_1,d_1),\ldots (Y_h,d_h) \}$ be a set of metric spaces. Let $\pi$ be a distribution on $\mathcal Y$. The original metric space $(X,d)$ embeds into $\pi$ with distortion $D$ if there is an $r>0$ such that for all $x,y\in X$, +\[r\leq \frac{\E_{i\from \pi} [d_i(\alpha_i(x),\alpha_i(y))]}{d(x,y)}\leq Dr.\] SODA23 paper also embeds $(X,d)$ into distribution. @@ -27,7 +27,7 @@ SODA23 paper also embeds $(X,d)$ into distribution. Consider the problem of embedding some finite metric into a tree metric. We can get an $O(n)$ lowerbound via the unit edge length cycle $C_n$. However, if embedding into distortions is allowed, we can do $O(\log n)$. \begin{theorem}[Bartal] -Let $(X,d)$ be a metric on $n$ points with diameter $\Delta$, let $\mathcal D T$ be the set of tree metrics that dominate $d$, there is a distribution $\pi$ on $\mathcal D T$ such that $(X,d)$ embeds into $pi$ with distortion $O(\log n\log \Delta)$. +Let $(X,d)$ be a metric space on $n$ points with diameter $\Delta$, let $\mathcal D T$ be the set of tree metrics that dominate $d$, there is a distribution $\pi$ on $\mathcal D T$ such that $(X,d)$ embeds into $pi$ with distortion $O(\log n\log \Delta)$. \end{theorem}