The complexity and decidability of separation
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.
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.
There is an O(n log2n) time algorithm to decide whether or not a puzzle is one-separable.
It is decidable whether or not a puzzle is separable.
Deciding separability is NP-hard.
There are puzzles which require time exponential in the number of edges to separate them.
KeywordsMove Sequence Segment Tree Computer Science Technical Report Proper Ancestor Relation Trap
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- [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
- [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
- [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
- [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
- [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
- [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
- [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
- [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
- [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
- [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
- [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
- [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