How to sort by walking on a tree

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 ∈ Sn. 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 O(n2) 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 NP-complete.