**Reference**:- Noga Alon, Mihai Bădoiu, Erik D. Demaine, Martin Farach-Colton, MohammadTaghi Hajiaghayi, and Anastasios Sidiropoulos, “Ordinal Embeddings of Minimum Relaxation: General Properties, Trees, and Ultrametrics”,
*ACM Transactions on Algorithms*, volume 4, number 4, August 2008, Article 46. **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 capture the ordinal behavior of ultrametrics and shortest-path metrics of unweighted trees. **Comments**:- This paper is also available from the ACM Digital Library.
**Length**:- The paper is 21 pages.
**Availability**:- The paper is available in PostScript (530k), gzipped PostScript (224k), and PDF (286k).
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**Related papers**:- Ordinal_APPROX2008 (Ordinal Embedding: Approximation Algorithms and Dimensionality Reduction)
- Ordinal_SODA2005 (Ordinal Embeddings of Minimum Relaxation: General Properties, Trees, and Ultrametrics)

See also other papers by Erik Demaine.

Last updated January 4, 2022 by Erik Demaine.