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

N1 - Funding Information:
We thank anonymous referees of the preliminary version [9] and of this journal version for their helpful suggestions. This work is supported in part by NSF grant CCF-1161626 and DARPA/AFOSR grant FA9550-12-1-0423 , and by MEXT/JSPS KAKENHI 24106004 , 25104521 , 25330003 , 25730003 , 26330009 and 15H00849 .

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.

KW - Combinatorial reconfiguration

KW - Graph algorithm

KW - Independent set

KW - Sliding token

KW - Tree

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U2 - 10.1016/j.tcs.2015.07.037

DO - 10.1016/j.tcs.2015.07.037

M3 - Article

AN - SCOPUS:84941260327

VL - 600

SP - 132

EP - 142

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -