Linear-Time Algorithm for Sliding Tokens on Trees
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...
Main Authors: | , , , , , , , , |
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Other Authors: | , |
Format: | Article |
Language: | English |
Published: |
Springer-Verlag,
2015-11-23T14:48:03Z.
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Subjects: | |
Online Access: | Get fulltext |
Summary: | 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. National Science Foundation (U.S.) (Grant CCF-1161626) United States. Defense Advanced Research Projects Agency (United States. Air Force Office of Scientific Research Grant FA9550-12-1-0423) |
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