Abstract
We consider algorithms for a simple one-dimensional point placement problem: given N points on a line, and noisy measurements of the distances be- tween many pairs of them, estimate the relative positions of the points. Problems of this flavor arise in a variety of contexts. The particular motivating example that inspired this work comes from molecular biology; the points are markers on a chromosome and the goal is to map their positions. The problem is NP-hard under reasonable assumptions. We present two algorithms for computing least squares estimates of the ordering and positions of the markers: a branch and bound al- gorithm and a highly effective heuristic search algorithm. The branch and bound algorithm is able to solve to optimality problems of 18 markers in about an hour, visiting about 106 nodes out of a search space of 1016 nodes. The local search algorithm usually was able to find the global minimum of problems of similar size in about one second, and should comfortably handle much larger problem instances.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Timothy Bishop. Linkage analysis: Progress and problems. Phil. Trans. R. Soc. Lond., 344:337–343, 1994.
Michael Boehnke, Kenneth Lange, and David Cox. Statistical methods for multipoint radiation hybrid mapping. Am. J. Hum. Genet., 49:1174–1188, 1991.
Kenneth H. Buetow and Aravinda Chakravarti. Multipoint gene mapping using seriation. I. General methods. Am. J. Hum. Genet., 41:180–188, 1987.
Donald Goldfarb and Shucheng Liu. An O(n 3 L) primal-dual potential reduction algorithm for solving convex quadratic programs. Mathematical Programming, 61:161–170, 1993.
Brendan Marshall Mumey. A fast heuristic algorithm for a probe mapping problem. In Proceedings of the Fifth International Conference on Intelligent Systems for Molecular Biology, pages 191–197, 1997.
Brendan Marshall Mumey. Some Computational Problems from Genomic Mapping. PhD thesis, Department of Computer Science and Engineering, University of Washington, 1997.
William R. Newell, Richard Mott, S. Beck, and Hans Lehrach. Construction of genetic maps using distance geometry. Genomics, 30:59–70, 1995.
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian Flannery. Numerical Recipes in C. Cambridge University Press, 1992.
Joshua Redstone and Walter L. Ruzzo. Algorithms for ordering DNA probes on chromosomes. Technical Report UW-CSE-98-12-04, Department of Computer Science and Engineering, University of Washington, December 1998.
Ger van den Engh, Ranier Sachs, and Barbara J. Trask. Estimating genomic distance from DNA sequence location in cell nuclei by a random walk model. Science, 257:1410–1412, 4 September 1992.
Harry Yeung and Walter L. Ruzzo. Algorithms for determining DNA sequence on chromosomes. Unpublished, March 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Redstone, J., Ruzzo, W.L. (2000). Algorithms for a Simple Point Placement Problem. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds) Algorithms and Complexity. CIAC 2000. Lecture Notes in Computer Science, vol 1767. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46521-9_3
Download citation
DOI: https://doi.org/10.1007/3-540-46521-9_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67159-6
Online ISBN: 978-3-540-46521-8
eBook Packages: Springer Book Archive