TY - JOUR

T1 - A linear-space algorithm for distance preserving graph embedding

AU - Asano, Tetsuo

AU - Bose, Prosenjit

AU - Carmi, Paz

AU - Maheshwari, Anil

AU - Shu, Chang

AU - Smid, Michiel

AU - Wuhrer, Stefanie

N1 - Funding Information:
✩ A preliminary version of this work appeared at CCCG 2007 [Tetsuo Asano, Prosenjit Bose, Paz Carmi, Anil Maheshwari, Chang Shu, Michiel Smid, Stefanie Wuhrer, Linear-space algorithm for distance preserving graph embedding with applications, in: Proceedings of the 19th Canadian Conference on Computational Geometry, 2007, pp. 185–188]. Research supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research on Priority Areas, Scientific Research (B), HPCVL, NSERC, NRC, MITACS, and MRI. We thank attendees of the 9th Korean Workshop on CG and Geometric Networks 2006. Work by T.A. was done in 2006 while visiting MPI, Carleton University, and NYU. We wish to thank anonymous referees for helpful comments. * Corresponding author at: Carleton University, Ottawa, Canada. E-mail address: swuhrer@scs.carleton.ca (S. Wuhrer).

PY - 2009/5/1

Y1 - 2009/5/1

N2 - The distance preserving graph embedding problem is to embed the vertices of a given weighted graph onto points in d-dimensional Euclidean space for a constant d such that for each edge the distance between their corresponding endpoints is as close to the weight of the edge as possible. If the given graph is complete, that is, if the weights are given as a full matrix, then multi-dimensional scaling [Trevor Cox, Michael Cox, Multidimensional Scaling, second ed., Chapman & Hall CRC, 2001] can minimize the sum of squared embedding errors in quadratic time. A serious disadvantage of this approach is its quadratic space requirement. In this paper we develop a linear-space algorithm for this problem for the case when the weight of any edge can be computed in constant time. A key idea is to partition a set of n objects into O(n) disjoint subsets (clusters) of size O(n) such that the minimum inter cluster distance is maximized among all possible such partitions. Experimental results are included comparing the performance of the newly developed approach to the performance of the well-established least-squares multi-dimensional scaling approach [Trevor Cox, Michael Cox, Multidimensional Scaling, second ed., Chapman & Hall CRC, 2001] using three different applications. Although least-squares multi-dimensional scaling gave slightly more accurate results than our newly developed approach, least-squares multi-dimensional scaling ran out of memory for data sets larger than 15000 vertices.

AB - The distance preserving graph embedding problem is to embed the vertices of a given weighted graph onto points in d-dimensional Euclidean space for a constant d such that for each edge the distance between their corresponding endpoints is as close to the weight of the edge as possible. If the given graph is complete, that is, if the weights are given as a full matrix, then multi-dimensional scaling [Trevor Cox, Michael Cox, Multidimensional Scaling, second ed., Chapman & Hall CRC, 2001] can minimize the sum of squared embedding errors in quadratic time. A serious disadvantage of this approach is its quadratic space requirement. In this paper we develop a linear-space algorithm for this problem for the case when the weight of any edge can be computed in constant time. A key idea is to partition a set of n objects into O(n) disjoint subsets (clusters) of size O(n) such that the minimum inter cluster distance is maximized among all possible such partitions. Experimental results are included comparing the performance of the newly developed approach to the performance of the well-established least-squares multi-dimensional scaling approach [Trevor Cox, Michael Cox, Multidimensional Scaling, second ed., Chapman & Hall CRC, 2001] using three different applications. Although least-squares multi-dimensional scaling gave slightly more accurate results than our newly developed approach, least-squares multi-dimensional scaling ran out of memory for data sets larger than 15000 vertices.

KW - Clustering

KW - Graph embedding

KW - Multi-dimensional scaling

UR - http://www.scopus.com/inward/record.url?scp=84867964613&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2008.06.004

DO - 10.1016/j.comgeo.2008.06.004

M3 - Article

AN - SCOPUS:84867964613

VL - 42

SP - 289

EP - 304

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 4

ER -