 
Summary: Ordinal Embeddings of Minimum Relaxation:
General Properties, Trees, and Ultrametrics
Noga Alon
Mihai Badoiu
Erik D. Demaine§
Martin FarachColton¶
MohammadTaghi Hajiaghayi
Anastasios Sidiropoulos
Abstract
We introduce a new notion of embedding, called minimumrelaxation ordinal embedding,
parallel to the standard notion of minimumdistortion (metric) embedding. In an ordinal em
bedding, it is the relative order between pairs of distances, and not the distances themselves,
that must be preserved as much as possible. The (multiplicative) relaxation of an ordinal em
bedding is the maximum ratio between two distances whose relative order is inverted by the
embedding. We develop several worstcase bounds and approximation algorithms on ordinal
embedding. In particular, we establish that ordinal embedding has many qualitative differences
from metric embedding, and capture the ordinal behavior of ultrametrics and shortestpath
metrics of unweighted trees.
1 Introduction
The classical field of multidimensional scaling (MDS) has witnessed a surge of interest in recent
