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An r-Dimensional Quadratic Placement Algorithm

Author

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  • Kenneth M. Hall

    (Department of General Services, State of California)

Abstract

In this paper the solution to the problem of placing n connected points (or nodes) in r-dimensional Euclidean space is given. The criterion for optimality is minimizing a weighted sum of squared distances between the points subject to quadratic constraints of the form X'X - 1, for each of the r unknown coordinate vectors. It is proved that the problem reduces to the minimization of a sum or r positive semi-definite quadratic forms which, under the quadratic constraints, reduces to the problem of finding r eigenvectors of a special "disconnection" matrix. It is shown, by example, how this can serve as a basis for cluster identification.

Suggested Citation

  • Kenneth M. Hall, 1970. "An r-Dimensional Quadratic Placement Algorithm," Management Science, INFORMS, vol. 17(3), pages 219-229, November.
    • Handle: RePEc:inm:ormnsc:v:17:y:1970:i:3:p:219-229
    DOI: 10.1287/mnsc.17.3.219
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    Cited by:

    1. Dorothy Kucar & Anthony Vannelli, 2006. "Using Eigenvectors to Partition Circuits," INFORMS Journal on Computing, INFORMS, vol. 18(2), pages 197-208, May.
    2. Ravi Kumar, K. & Hadjinicola, George C. & Lin, Ting-li, 1995. "A heuristic procedure for the single-row facility layout problem," European Journal of Operational Research, Elsevier, vol. 87(1), pages 65-73, November.
    3. Gautier M Krings & Jean-François Carpantier & Jean-Charles Delvenne, 2014. "Trade Integration and Trade Imbalances in the European Union: A Network Perspective," PLOS ONE, Public Library of Science, vol. 9(1), pages 1-14, January.
    4. Laleh Behjat & Dorothy Kucar & Anthony Vannelli, 2002. "A Novel Eigenvector Technique for Large Scale Combinatorial Problems in VLSI Layout," Journal of Combinatorial Optimization, Springer, vol. 6(3), pages 271-286, September.
    5. Philipp Hungerländer & Franz Rendl, 2013. "A computational study and survey of methods for the single-row facility layout problem," Computational Optimization and Applications, Springer, vol. 55(1), pages 1-20, May.
    6. Amaral, Andre R.S., 2006. "On the exact solution of a facility layout problem," European Journal of Operational Research, Elsevier, vol. 173(2), pages 508-518, September.
    7. Nascimento, Mariá C.V. & de Carvalho, André C.P.L.F., 2011. "Spectral methods for graph clustering - A survey," European Journal of Operational Research, Elsevier, vol. 211(2), pages 221-231, June.
    8. André R. S. Amaral, 2008. "An Exact Approach to the One-Dimensional Facility Layout Problem," Operations Research, INFORMS, vol. 56(4), pages 1026-1033, August.
    9. Jintae Park & Sungha Yoon & Chaeyoung Lee & Junseok Kim, 2020. "A Simple Method for Network Visualization," Mathematics, MDPI, vol. 8(6), pages 1-13, June.
    10. Dorit S. Hochbaum, 2013. "A Polynomial Time Algorithm for Rayleigh Ratio on Discrete Variables: Replacing Spectral Techniques for Expander Ratio, Normalized Cut, and Cheeger Constant," Operations Research, INFORMS, vol. 61(1), pages 184-198, February.
    11. Bhupatiraju, Samyukta & Nomaler, Önder & Triulzi, Giorgio & Verspagen, Bart, 2012. "Knowledge flows – Analyzing the core literature of innovation, entrepreneurship and science and technology studies," Research Policy, Elsevier, vol. 41(7), pages 1205-1218.

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