The complexity and decidability of separation

  • Bernard Chazelle
  • Thomas Ottmann
  • Eljas Soisalon-Soininen
  • Derick Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 172)


We study the difficulty of solving instances of a new family of sliding block puzzles called SEPARATIONTM. Each puzzle in the family consists of an arrangement in the plane of n rectilinear wooden blocks, n > 0. The aim is to discover a sequence of rectilinear moves which when carried out will separate each piece to infinity. If there is such a sequence of moves we say the puzzle or arrangement is separable and if each piece is moved only once we say it is one-separable. Furthermore if it is one-separable with all moves being in the same direction we say it is iso-separable.

We prove:
  1. (1)

    There is an O(n log n) time algorithm to decide whether or not a puzzle is iso-separable, where the blocks have a total of n edges.

  2. (2)

    There is an O(n log2n) time algorithm to decide whether or not a puzzle is one-separable.

  3. (3)

    It is decidable whether or not a puzzle is separable.

  4. (4)

    Deciding separability is NP-hard.

  5. (5)

    There are puzzles which require time exponential in the number of edges to separate them.



Move Sequence Segment Tree Computer Science Technical Report Proper Ancestor Relation Trap 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BCR]
    Baker, B.S., Coffman Jr., E.G., and Rivest, R.L., Orthogonal Packings in Two Dimensions, SIAM Journal on Computing 9 (1980), 846–855.Google Scholar
  2. [BW]
    Bentley, J.L., and Wood, D., An Optimal Worst-Case Algorithm for Reporting Intersections of Rectangles, IEEE Transactions on Computers, C-29 (1980), 571–577.Google Scholar
  3. [E]
    Edelsbrunner, H., A Time-and Space-Optimal Solution for the Planar All Intersecting Rectangles Problem. Tech. Rep., University of Graz, IIG Rep. 50, April 1980.Google Scholar
  4. [GY]
    Guibas, L.J., and Yao, F.F., On Translating a Set of Rectangles, Proceedings of the Tenth Annual ACM-SIGACT Symposium on Theory of Computing (1980), 154–160.Google Scholar
  5. [HJW]
    Hopcroft, J.E., Joseph, D.A., and Whitesides, S.H., On the Movement of Robot Arms in 2-Dimensional Bounded Regions, Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, (1982), 280–289.Google Scholar
  6. [LPW]
    Lozano-Perez, T., and Wesley, M., An Algorithm for Planning Collision-Free Paths among Polyhedral Obstacles, Communications of the ACM 22 (1979), 560–570.Google Scholar
  7. [OSC]
    O'Dúnlaing, C., Sharir, M., and Yap, C.K., Retraction: A New Approach to Motion Planning, Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (1983), 207–220.Google Scholar
  8. [R]
    Reif, J., Complexity of the Mover's Problem and Generalizations, Proceeding of the 20th Annual Symposium on Foundations of Computer Science (1979), 421–427.Google Scholar
  9. [SLW]
    Schlag, M., Liao, Y.Z., and Wong, C.K., An Algorithm for Optimal Two-Dimensional Compaction Layouts, IBM Research Center, Yorktown, Research Report RC 9739, 1982.Google Scholar
  10. [SS1]
    Schwartz, J.T., and Sharir, M., On the Piano Mover's Problem: I. The Special Case of a Rigid Polygonal Body Moving amidst Polygonal Barriers, Communications on Pure and Applied Mathematics (1983), to appear.Google Scholar
  11. [SS2]
    Schwartz, J.T., and Sharir, M., On the Piano Mover's Problem: II. General Techniques for Computing Topological Properties of Real Alagebraic Manifolds, Advances in Applied Mathematics (1983), to appear.Google Scholar
  12. [SS3]
    Schwartz, J.T., and Sharir, M., On the Piano Mover's Problem: III. Coordinating the Motion of Several Independent Bodies: The Special Case of Circular Bodies Moving amidst Polygonal Barriers, New York University Courant Institute Computer Science Technical Report, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Bernard Chazelle
    • 1
  • Thomas Ottmann
    • 2
  • Eljas Soisalon-Soininen
    • 3
  • Derick Wood
    • 4
  1. 1.Computer Science DepartmentBrown UniversityProvidenceUSA
  2. 2.Institut für Angewandte Informatik und Formale BeschreibungsverfahrenUniversität KarlsruheKarlsruheW. Germany
  3. 3.Department of Computer ScienceUniversity of HelsinkiHelsinki 25Finland
  4. 4.Data Structuring Group, Department of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations