Ordinal embeddings of minimum relaxation: General properties, trees, and ultrametrics

Authors:
Noga Alon;Mihai Bădoiu;Erik D. Demaine;Martin Farach-Colton;Mohammadtaghi Hajiaghayi;Anastasios Sidiropoulos
Affiliations:
Tel Aviv University, Tel Aviv, Israel;Google Inc., NY, USA;MIT, Cambridge, MA, USA;Rutgers University, Piscataway, NJ, USA;AT&T Labs——Research, Florham Park, NJ, USA;MIT, Florham Park, Cambridge, MA, USA
Venue:
ACM Transactions on Algorithms (TALG)
Year:
2008

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Abstract

We introduce a new notion of embedding, called minimum-relaxation ordinal embedding, parallel to the standard notion of minimum-distortion (metric) embedding. In an ordinal embedding, 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 embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop several worst-case bounds and approximation algorithms on ordinal embedding. In particular, we establish that ordinal embedding has many qualitative differences from metric embedding, and we capture the ordinal behavior of ultrametrics and shortest-path metrics of unweighted trees.