### Abstract

Suppose that we are given two independent sets I_{b} and I_{r} of a graph such that |Ib|=|Ir|, and imagine that a token is placed on each vertex in I_{b}. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I_{b} into I_{r} so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I_{b} and I_{r} whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

Original language | English |
---|---|

Pages (from-to) | 132-142 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 600 |

DOIs | |

Publication status | Published - Oct 4 2015 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*600*, 132-142. https://doi.org/10.1016/j.tcs.2015.07.037

**Linear-time algorithm for sliding tokens on trees.** / Demaine, Erik D.; Demaine, Martin L.; Fox-Epstein, Eli; Hoang, Duc A.; Ito, Takehiro; Ono, Hirotaka; Otachi, Yota; Uehara, Ryuhei; Yamada, Takeshi.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 600, pp. 132-142. https://doi.org/10.1016/j.tcs.2015.07.037

}

TY - JOUR

T1 - Linear-time algorithm for sliding tokens on trees

AU - Demaine, Erik D.

AU - Demaine, Martin L.

AU - Fox-Epstein, Eli

AU - Hoang, Duc A.

AU - Ito, Takehiro

AU - Ono, Hirotaka

AU - Otachi, Yota

AU - Uehara, Ryuhei

AU - Yamada, Takeshi

PY - 2015/10/4

Y1 - 2015/10/4

N2 - Suppose that we are given two independent sets Ib and Ir of a graph such that |Ib|=|Ir|, and imagine that a token is placed on each vertex in Ib. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms Ib into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between Ib and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

AB - Suppose that we are given two independent sets Ib and Ir of a graph such that |Ib|=|Ir|, and imagine that a token is placed on each vertex in Ib. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms Ib into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between Ib and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

UR - http://www.scopus.com/inward/record.url?scp=84941260327&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941260327&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2015.07.037

DO - 10.1016/j.tcs.2015.07.037

M3 - Article

VL - 600

SP - 132

EP - 142

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -