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International Conference on Computational Science and Its Applications

ICCSA 2017: Computational Science and Its Applications – ICCSA 2017 pp 494–508Cite as

A Distance Matrix Completion Approach to 1-Round Algorithms for Point Placement in the Plane

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10405))

Abstract

In this paper we propose a 1-round algorithm for point placement in the plane in an adversarial model. The distance query graph presented to the adversary is chordal. The remaining distances are uniquely determined using a distance matrix completion algorithm for chordal graphs, based on a result by Bakonyi and Johnson [4].

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References

  1. Alam, M.S., Mukhopadhyay, A.: More on generalized jewels and the point placement problem. J. Graph Algorithms Appl. 18(1), 133–173 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alam, M.S., Mukhopadhyay, A.: Three paths to point placement. In: Ganguly, S., Krishnamurti, R. (eds.) CALDAM 2015. LNCS, vol. 8959, pp. 33–44. Springer, Cham (2015). doi:10.1007/978-3-319-14974-5_4

    Google Scholar 

  3. Aspnes, J., Eren, T., Goldenberg, D.K., Stephen Morse, A., Whiteley, W., Yang, Y.R., Anderson, B.D.O., Belhumeur, P.N.: A theory of network localization. IEEE Trans. Mob. Comput. 5(12), 1663–1678 (2006)

    Article  Google Scholar 

  4. Bakonyi, M., Johnson, C.R.: The euclidian distance matrix completion problem. SIAM J. Matrix Anal. Appl. 16(2), 646–654 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biswas, P., Liang, T.-C., Wang, T.-C., Ye, Y.: Semidefinite programming based algorithms for sensor network localization. TOSN 2(2), 188–220 (2006)

    Article  Google Scholar 

  6. Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)

    Article  MATH  Google Scholar 

  7. Chin, F.Y.L., Leung, H.C.M., Sung, W.K., Yiu, S.M.: The point placement problem on a line – improved bounds for pairwise distance queries. In: Giancarlo, R., Hannenhalli, S. (eds.) WABI 2007. LNCS, vol. 4645, pp. 372–382. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74126-8_35

    Chapter  Google Scholar 

  8. Damaschke, P.: Point placement on the line by distance data. Discrete Appl. Math. 127(1), 53–62 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Damaschke, P.: Randomized vs. deterministic distance query strategies for point location on the line. Discrete Appl. Math. 154(3), 478–484 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ellis, R.L., Lay, D.: Rank-preserving extensions of band matrices. Linear Multilinear Algebra 26, 147–179 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, vol. 57). North-Holland Publishing Co., Amsterdam (2004)

    Google Scholar 

  12. Grone, R., Johnson, C.R., Sa, E.M., Wolkowicz, H.: Positive definite completions of partial hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Markenzon, L., Vernet, O., Araujo, L.H.: Two methods for the generation of chordal graphs. Annals OR 157(1), 47–60 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mukhopadhyay, A., Sarker, P.K., Kannan, K.K.V.: Point placement algorithms: an experimental study. Int. J. Exp. Algorithms 6(1), 1–13 (2016)

    Google Scholar 

  15. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schoenberg, I.J.: Remarks to Maurice Frechet’s article “sur la definition axiomatique d’une classe d’espace distancis vectoriellement applicable sur l’espace de hilbert”. Ann. Math. 36(3), 724–731 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  17. Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3(1), 19–22 (1938)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Computer Science, University of Windsor, Windsor, ON, N9B 3P4, Canada

    Md. Zamilur Rahman, Asish Mukhopadhyay & Cory Jean

  2. Odette School of Business, University of Windsor, Windsor, ON, N9B 3P4, Canada

    Yash Pal Aneja

Authors
  1. Md. Zamilur Rahman
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  2. Asish Mukhopadhyay
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  3. Yash Pal Aneja
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  4. Cory Jean
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Corresponding author

Correspondence to Asish Mukhopadhyay .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Trieste, Trieste, Italy

    Giuseppe Borruso

  5. Polytechnic University of Bari, Bari, Italy

    Carmelo M. Torre

  6. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  7. Monash University, Clayton, Victoria, Australia

    David Taniar

  8. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  9. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  10. University of Trieste, Trieste, Italy

    Alfredo Cuzzocrea

A Appendix

A Appendix

Below we show two more examples where the input chordal graphs are automatically generated by our software. Also shown are the incomplete distance matrices and the coordinates of the points obtained from their corresponding matrix completions (Tables 1, 2, 3 and 4).

Fig. 10.
figure 10

A chordal graph with 10 vertices

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Table 1. Distance matrix corresponding to the chordal graph of Fig. 10
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Table 2. x and y coordinates of points obtained from the matrix completion of the chordal graph of Fig. 10
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Fig. 11.
figure 11

A chordal graph with 15 vertices

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Table 3. Distance matrix corresponding to the chordal graph of Fig. 11
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Table 4. x and y coordinates of points obtained from the matrix completion of the chordal graph of Fig. 11
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Rahman, M.Z., Mukhopadhyay, A., Aneja, Y.P., Jean, C. (2017). A Distance Matrix Completion Approach to 1-Round Algorithms for Point Placement in the Plane. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10405. Springer, Cham. https://doi.org/10.1007/978-3-319-62395-5_34

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  • DOI: https://doi.org/10.1007/978-3-319-62395-5_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62394-8

  • Online ISBN: 978-3-319-62395-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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