Abstract
Consider a graph G with n vertices. On each vertex we place a box. These n vertices and n boxes are both numbered from 1 to n and initially shuffled according to a permutation π ∈ S n . We introduce a sorting problem for a single robot: In every step, the robot can walk along an edge of G and can carry at most one box at a time. At a vertex, it may swap the box placed there with the box it is carrying. How many steps does the robot need to sort all the boxes?
We present an algorithm that produces a shortest possible sorting walk for such a robot if G is a tree. The algorithm runs in time \(\mathcal{O}(n^2)\) and can be simplified further if G is a path. We show that for planar graphs the problem of finding a shortest possible sorting walk is \(\mathcal{NP}\)-complete.
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Graf, D. (2015). How to Sort by Walking on a Tree. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_54
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DOI: https://doi.org/10.1007/978-3-662-48350-3_54
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