# The complexity and decidability of separation

## 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*.

- (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)
There is an O(

*n*log^{2}*n*) time algorithm to decide whether or not a puzzle is one-separable. - (3)
It is decidable whether or not a puzzle is separable.

- (4)
Deciding separability is

*NP*-hard. - (5)
There are puzzles which require time exponential in the number of edges to separate them.

## Keywords

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

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