Linear-Time Algorithm for Sliding Tokens on Trees
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Demaine_Polynomial-time.pdf
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Author(s)
Demaine, Erik D.
Demaine, Martin L.
Fox-Epstein, Eli
Hoang, Duc A.
Ito, Takehiro
Ono, Hirotaka
Otachi, Yota
Uehara, Ryuhei
Yamada, Takeshi
Alternative Title
Polynomial-Time Algorithm for Sliding Tokens on Trees
Date Issued
2014
Journal
Algorithms and Computation
Publisher
Springer-Verlag
Citation
Demaine, Erik D., Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. “Polynomial-Time Algorithm for Sliding Tokens on Trees.” Lecture Notes in Computer Science (2014): 389–400.
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1007/978-3-319-13075-0_31
