# 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

Research output: Contribution to journalArticle

32 Citations (Scopus)

### Abstract

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.

Original language English 132-142 11 Theoretical Computer Science 600 https://doi.org/10.1016/j.tcs.2015.07.037 Published - Oct 4 2015

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computer Science(all)

• ## Cite this

Demaine, E. D., Demaine, M. L., Fox-Epstein, E., Hoang, D. A., Ito, T., Ono, H., Otachi, Y., Uehara, R., & Yamada, T. (2015). Linear-time algorithm for sliding tokens on trees. Theoretical Computer Science, 600, 132-142. https://doi.org/10.1016/j.tcs.2015.07.037