Constant Factor Time Optimal Multi-Robot Routing on High-Dimensional Grids in Mostly Sub-Quadratic Time

Let be an grid. Assuming that each is occupied by a robot and a robot may move to a neighboring vertex in a step via synchronized rotations along cycles of , we first establish that the arbitrary reconfiguration of labeled robots on can be performed in makespan and requires running time in the worst case and when is non-degenerate (i.e., nearly one dimensional). The resulting algorithm, \isag, provides average case -approximate (i.e., constant factor) time optimality guarantee. In the case when all dimensions are of similar size , the running time of \isag approaches a linear . Define as the largest distance between individual initial and goal configurations over all robots for a given problem instance , building on \isag, we develop the \pafalgo (\paf) algorithm that computes makespan solutions for using mostly running time. \paf provides worst case -approximation regarding solution time optimality. We note that the worst case running time for the problem is .
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