  Given metric spaces $(X,d)$ and $(Y,\rho)$ and an ordering
$x_1,x_2,\ldots,x_n$ of $(X,d)$, an embedding $f: X \rightarrow Y$ is said to
have a prioritized distortion $\alpha(\cdot)$, if for any pair $x_j,x'$ of
distinct points in $X$, the distortion provided by $f$ for this pair is at most
$\alpha(j)$. If $Y$ is a normed space, the embedding is said to have
prioritized dimension $\beta(\cdot)$, if $f(x_j)$ may have nonzero entries only
in its first $\beta(j)$ coordinates.
  The notion of prioritized embedding was introduced by \cite{EFN15}, where a
general methodology for constructing such embeddings was developed. Though this
methodology enables \cite{EFN15} to come up with many prioritized embeddings,
it typically incurs some loss in the distortion. This loss is problematic for
isometric embeddings. It is also troublesome for Matousek's embedding of
general metrics into $\ell_\infty$, which for a parameter $k = 1,2,\ldots$,
provides distortion $2k-1$ and dimension $O(k \log n \cdot n^{1/k})$.
  In this paper we devise two lossless prioritized embeddings. The first one is
an isometric prioritized embedding of tree metrics into $\ell_\infty$ with
dimension $O(\log j)$. The second one is a prioritized Matousek's embedding of
general metrics into $\ell_\infty$, which provides prioritized distortion $2
\lceil k {{\log j} \over {\log n}} \rceil - 1$ and dimension $O(k \log n \cdot
n^{1/k})$, again matching the worst-case guarantee $2k-1$ in the distortion of
the classical Matousek's embedding. We also provide a dimension-prioritized
variant of Matousek's embedding. Finally, we devise prioritized embeddings of
general metrics into (single) ultra-metric and of general graphs into (single)
spanning tree with asymptotically optimal distortion.
