Approximation algorithms for low-distortion embeddings into low-dimensional spaces

We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. We give an O([square root] n)-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. For the same problem, we give an exact algorithm, with running-time exponential in the distortion. We complement these results by showing that the problem is NP-hard to [alpha]-approximate, for some constant [alpha] > 1. For the two-dimensional case, we show a O([square root] n) upper bound for the distortion required to embed an n-point subset of the two-dimensional sphere, into the plane. We prove that this bound is asymptotically tight, by exhibiting n-point subsets such that any embedding into the plane has distortion [omega]([square root] n). These techniques yield a O(1)-approximation algorithm for the problem of embedding an n-point subset of the sphere into the plane.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2005

Author dc:creator

• Sidiropoulos, Anastasios

Advisor dc:contributor.advisor

• Piotr Indyk.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/34126

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/34126