TITLE : Metric Embeddings with Relaxed Guarantees
AUTHORS : Ittai Abraham, Yair Bartal, T-H. Hubert Chan, Kedar Dhamdhere, Anupam Gupta, Jon Kleinberg, Ofer Neiman, Aleksandrs Slivkins
ABSTRACT:
We consider the problem of embedding finite metrics with {\em slack}:
we seek to produce embeddings with small dimension and distortion
while allowing a (small) constant fraction of all distances to be
arbitrarily distorted. This definition is motivated by recent research
in the networking community, which achieved striking empirical success
at embedding Internet latencies with low distortion into
low-dimensional Euclidean space, provided that some small slack is
allowed.
Answering an open question of Kleinberg, Slivkins, and Wexler
\cite{KSW04}, we show that provable guarantees of this type can in
fact be achieved in general: any finite metric can be embedded, with
constant slack and constant distortion, into constant-dimensional
Euclidean space. We then show that there exist stronger embeddings
into $\ell_1$ which exhibit {\em gracefully degrading} distortion:
these is a single embedding into $\ell_1$ that achieves distortion at
most $O(\log\freps)$ on all but at most an $\e$ fraction of distances,
{\em simultaneously} for all $\e > 0$. We extend this with distortion
$O(\log\freps)^{1/p}$ to maps into general $\ell_p$, $p\geq 1$ for
several classes of metrics, including those with bounded doubling
dimension and those arising from the shortest-path metric of a graph
with an excluded minor. Finally, we show that many of our
constructions are tight, and give a general technique to obtain lower
bounds for $\e$-slack embeddings from lower bounds for low-distortion
embeddings.