Abstract
We study the difficulty of solving instances of a new family of sliding block puzzles called SEPARATION TM. 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)
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.
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(2)
There is an O(n log2 n) time algorithm to decide whether or not a puzzle is one-separable.
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(3)
It is decidable whether or not a puzzle is separable.
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(4)
Deciding separability is NP-hard.
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(5)
There are puzzles which require time exponential in the number of edges to separate them.
The work of the first author was supported in part by the Office of Naval Research and the Defense Advanced Research Projects Agency under Contract N00014-83-K-0146 and ARPA Order No. 4786 and under a National Science Foundation Grant No. MCS-8303925, that of the third by a grant from the Alexander von Humboldt Foundation, and that of the fourth by a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692.
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© 1984 Springer-Verlag Berlin Heidelberg
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Chazelle, B., Ottmann, T., Soisalon-Soininen, E., Wood, D. (1984). The complexity and decidability of separation. In: Paredaens, J. (eds) Automata, Languages and Programming. ICALP 1984. Lecture Notes in Computer Science, vol 172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-13345-3_10
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DOI: https://doi.org/10.1007/3-540-13345-3_10
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