Abstract
The point placement problem is to determine the locations of n distinct points on a line uniquely (up to translation and reflection) by making the fewest possible pairwise distance queries of an adversary (an adversary is just a source of true distances). A number of deterministic and randomized algorithms are available when distances are known exactly. In this paper, we discuss the problem in an inexact model. This is when distances returned by the adversary are not exact; instead, only upper and lower bounds on the distances are provided. We explore three different approaches to this problem. The first is based on an adaption of a distance geometry approach that Havel and Crippen [6] used to solve the molecular conformation problem. The second is based on a linear programming solution to a set of difference constraints that was used by Mumey [7] to solve a probe location problem arsing in DNA sequence analysis. The third is based on a heuristic called Stochastic Proximity Embedding, proposed by Agrafiotis [8]. Extensive experiments were carried out to determine the most promising approach vis-a-vis two parameters: run-time and quality of the embedding, measured by computing a certain stress function.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Blumenthal, L.M.: Theory and Applications of Distance Geometry (1970)
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). https://doi.org/10.1007/978-3-540-74126-8_35
Damaschke, P.: Point placement on the line by distance data. Discrete Appl. Math. 127(1), 53–62 (2003)
Malliavin, T.E., Mucherino, A., Nilges, M.: Distance geometry in structural biology: new perspectives. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.) Distance Geometry, pp. 329–350. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-5128-0_16
Mukhopadhyay, A., Sarker, P.K., Kannan, K.K.V.: Randomized versus deterministic point placement algorithms: an experimental study. In: Gervasi, O., et al. (eds.) ICCSA 2015. LNCS, vol. 9156, pp. 185–196. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21407-8_14
Crippen, G.M., Havel, T.F.: Distance Geometry and Molecular Conformation, vol. 74. Research Studies Press, Somerset, England (1988)
Mumey, B.: Probe location in the presence of errors: a problem from dna mapping. Discrete Appl. Math. 104(1), 187–201 (2000)
Agrafiotis, D.K.: Stochastic proximity embedding. J. Comput. Chem. 24(10), 1215–1221 (2003)
Skiena, S.S., Smith, W.D., Lemke, P.: Reconstructing sets from interpoint distances (extended abstract). In: SCG 1990: Proceedings of the Sixth Annual Symposium on Computational Geometry, pp. 332–339. ACM, New York (1990)
Smith, H., Wilcox, K.W.: A restriction enzyme from hemophilus influenzae. I. Purification and general properties. J. Mol. Biol. 51, 379–391 (1970)
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). https://doi.org/10.1007/978-3-319-14974-5_4
Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3(1), 19–22 (1938)
Emiris, I.Z., Psarros, I.D.: Counting Euclidean embeddings of rigid graphs. CoRR abs/1402.1484 (2014)
Saxe, J.B.: Embeddability of weighted graphs in \(k\)-space is strongly NP-hard. In: 17th Allerton Conference on Communication, Control and Computing, pp. 480–489 (1979)
Mukhopadhyay, A., Rao, S.V., Pardeshi, S., Gundlapalli, S.: Linear layouts of weakly triangulated graphs. Discrete Math. Algorithms Appl. 8(3), 1–21 (2016)
Mukhopadhyay, A., Sarker, P.K., Kannan, K.K.: Point placement algorithms: an experimental study. Int. J. Exp. Algorithms 6(1), 1–13 (2016)
Havel, T.F.: Distance geometry: Theory, algorithms, and chemical applications. In: Encyclopedia of Computational Chemistry, vol. 120 (1998)
Newell, W.R., Mott, R., Beck, S., Lehrach, H.: Construction of genetic maps using distance geometry. Genomics 30(1), 59–70 (1995)
Dress, A.W.M., Havel, T.F.: Shortest-path problems and molecular conformation. Discrete Appl. Math. 19(1–3), 129–144 (1988)
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. The MIT Press and McGraw-Hill Book Company, Cambridge, New York (1989)
Buetow, K.H., Chakravarti, A.: Multipoint gene mapping using seriation. I. General methods. Am. J. Hum. Genet. 41(2), 180 (1987)
Pinkerton, B.: Results of a simulated annealing algorithm for fish mapping. Communicated by Dr. Larry Ruzzo, University of Washington (1993)
Redstone, J., Ruzzo, W.L.: Algorithms for a simple point placement problem. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 32–43. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46521-9_3
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer-Verlag GmbH Germany, part of Springer Nature
About this chapter
Cite this chapter
Kannan, K.K.V., Sarker, P.K., Turdaliev, A., Mukhopadhyay, A., Rahman, M.Z. (2019). A Study of Three Different Approaches to Point Placement on a Line in an Inexact Model. In: Gavrilova, M., Tan, C. (eds) Transactions on Computational Science XXXIV. Lecture Notes in Computer Science(), vol 11820. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59958-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-59958-7_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-59957-0
Online ISBN: 978-3-662-59958-7
eBook Packages: Computer ScienceComputer Science (R0)