2026-09-18T22:53:41Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/341262022-01-13T07:54:29Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Piotr Indyk.
Sidiropoulos, Anastasios
Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2006-09-28T15:05:46Z
2006-09-28T15:05:46Z
2005
2005
http://hdl.handle.net/1721.1/34126
67767097
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.
Includes bibliographical references (p. 33-35).
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.
by Anastasios Sidiropoulos.
S.M.
35 p.
1325523 bytes
1326855 bytes
application/pdf
application/pdf
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Electrical Engineering and Computer Science.
Approximation algorithms for low-distortion embeddings into low-dimensional spaces
Thesis
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<Abstract>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.</Abstract>
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