Advertisement

A New Dynamic Programming Algorithm for Orthogonal Ruler Folding Problem in d-Dimensional Space

  • Ali Nourollah
  • Mohammad Reza Razzazi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4705)

Abstract

A chain or n-link is a sequence of n links whose lengths are fixed joined together from their endpoints, free to turn about their endpoints, which act as joints. “Ruler Folding Problem”, which is NP-Complete is to find the minimum length of the folded chain in one dimensional space. The best result for ruler folding problem is reported by Hopcroft et al. in one dimensional space which requires O(nL 2) time complexity, where L is length of the longest link in the chain and links have integer value lengths. We propose a dynamic programming approach to fold a given chain whose links have integer lengths in a minimum length in O(nL) time and space. We show that by generalizing the algorithm it can be used in d-dimensional space for orthogonal ruler folding problem such that it requires O(2 d ndL d ) time using O(2 d ndL d ) space.

Keywords

Ruler Folding Problem Carpenter’s Ruler Dynamic Programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hopcroft, J., Joseph, D., Whitesides, S.: On the movement of robot arms in 2-dimensional bounded regions. SIAM J. Comput. 14(2), 315–333 (1985)zbMATHCrossRefGoogle Scholar
  2. 2.
    Whitesides, S.: Chain Reconfiguration. The Ins and Outs, Ups and Downs of Moving Polygons and Polygonal Linkages. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 1–13. Springer, Heidelberg (2001)Google Scholar
  3. 3.
    Calinescu, G., Dumitrescu, A.: The carpenter’s ruler folding problem. In: Goodman, J., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, pp. 155–166. Mathematical Sciences Research Institute Publications, Cambridge University Press (2005)Google Scholar
  4. 4.
    Kantabutra, V.: Reaching a point with an unanchored robot arm in a square. International journal of Computational Geometry & Applications 7(6), 539–549 (1997)zbMATHCrossRefGoogle Scholar
  5. 5.
    Biedl, T., Demaine, E., Demaine, M., Lazard, S., Lubiw, A., O’Rourke, J., Robbins, S., Streinu, I., Toussaint, G., Whitesides, S.: A note on reconfigurating tree linkages: Trees can lock. Discrete Appl. Math. (2001)Google Scholar
  6. 6.
    Biedl, T., Lubiw, A., Sun, J.: When Can a Net Fold to a Polyhedron? In: Eleventh Canadian Conference on Computational Geometry, U. British Columbia (1999)Google Scholar
  7. 7.
    Lenhart, W.J., Whitesides, S.: Reconfiguring Closed Polygonal Chains in Euclidean d-Space. Discrete and Computational Geometry 13, 123–140 (1995)zbMATHCrossRefGoogle Scholar
  8. 8.
    Whitesides, S.: Algorithmic issues in the geometry of planar linkage movement. Australian Computer Journal, Special Issue on Algorithms 24(2), 42–50 (1992)Google Scholar
  9. 9.
    O’Rourke, J.: Folding and unfolding in computational geometry. Discrete and Computational Geometry 1763, 258–266 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ali Nourollah
    • 1
  • Mohammad Reza Razzazi
    • 1
    • 2
  1. 1.Software Systems R&D Lab., Department of Computer Engineering & IT, Amirkabir University of Technology, #424 Hafez Avenue, P. O. Box 15875-4413, TehranIran
  2. 2.Institute for Studies in Theoretical Physics and Mathematics (I.P.M.) 

Personalised recommendations