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arXiv · 1406.6576

Linear-Time Algorithm for Sliding Tokens on Trees

Erik D. Demaine·Martin L. Demaine·Eli Fox-Epstein·Duc A. Hoang·Takehiro Ito·Hirotaka Ono·Yota Otachi·Ryuhei Uehara·Takeshi Yamada
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Abstract

Suppose that we are given two independent sets $I_b$ and $I_r$ of a graph such that $|I_b|=|I_r|$, 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.

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Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, Takeshi Yamada. 2014-06-25. Linear-Time Algorithm for Sliding Tokens on Trees. https://arxiv.org/abs/1406.6576

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