We design an algorithm to embed graph metrics into ℓp with dimension and distortion both dependent only upon the bandwidth of the graph. In particular, we show that any graph of bandwidth k embeds with distortion polynomial in k into ℓpO(logk), 1≤p≤. Prior to our result the only known embedding with distortion independent of n was into high dimensional ℓ1 and had distortion exponential in k. Our low dimensional embedding is based on a general method for reducing the dimension of an ℓp embedding. This method requires that the embedding satisfy certain conditions, and the dimension is reduced to the intrinsic dimension of the point set, without substantially increasing the distortion. We observe that the family of graphs with bounded bandwidth are doubling, thus our main result can be viewed as a positive answer to a conjecture of Assouad (1983) , limited to this family. We also study an extension to graphs of bounded tree-bandwidth.
Bibliographical noteFunding Information:
E-mail addresses: firstname.lastname@example.org (Y. Bartal), email@example.com (D.E. Carroll), firstname.lastname@example.org (A. Meyerson), email@example.com (O. Neiman). 1 The work was done in part while the author was at the Center for the Mathematics of Information, Caltech, and the Institute for Pure and Applied Mathematics, UCLA. Supported in part by a grant from the Israeli Science Foundation (195/02) and in part by a grant from the National Science Foundation (NSF CCF-065253). 2 Research done while a student at UCLA. 3 Research supported by the National Science Foundation under Grant No. CCF-106540. 4 Supported by ISF grant No. 523/12 and by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 303809.
- Low dimension
- Metric embedding