Linear-Time Algorithm for Sliding Tokens on Trees

• Main Authors: Demaine, Erik D. (Contributor), Demaine, Martin L. (Contributor), Fox-Epstein, Eli (Author), Hoang, Duc A. (Author), Ito, Takehiro (Author), Ono, Hirotaka (Author), Otachi, Yota (Author), Uehara, Ryuhei (Author), Yamada, Takeshi (Author)
• Other Authors: Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor), Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor)
• Format: Article
• Language: English
• Published: Springer-Verlag, 2015-11-23T14:48:03Z.
• Subjects: Article
• Online Access: Get fulltext

 Suppose that we are given two independent sets I [subscript b] and I [subscript r] of a graph such that ∣ I [subscript b] ∣ = ∣ I [subscript r] ∣, and imagine that a token is placed on each vertex in I [subscript b]. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I [subscript b] and I [subscript 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 show that the problem is solvable for trees in quadratic time. Our proof is constructive: for a yes-instance, we can find an actual sequence of independent sets between I [subscript b] and I [subscript r] whose length (i.e., the number of token-slides) is quadratic. We note that there exists an infinite family of instances on paths for which any sequence requires quadratic length.