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

Jingjin Yu
Abstract

Let G=(V,E)G = (V, E) be an m1××mkm_1 \times \ldots \times m_k grid. Assuming that each vVv \in V is occupied by a robot and a robot may move to a neighboring vertex in a step via synchronized rotations along cycles of GG, we first establish that the arbitrary reconfiguration of labeled robots on GG can be performed in O(kimi)O(k\sum_i m_i) makespan and requires O(V2)O(|V|^2) running time in the worst case and o(V2)o(|V|^2) when GG is non-degenerate (i.e., nearly one dimensional). The resulting algorithm, \isag, provides average case O(1)O(1)-approximate (i.e., constant factor) time optimality guarantee. In the case when all dimensions are of similar size O(V1k)O(|V|^{\frac{1}{k}}), the running time of \isag approaches a linear O(V)O(|V|). Define dg(p)d_g(p) as the largest distance between individual initial and goal configurations over all robots for a given problem instance pp, building on \isag, we develop the \pafalgo (\paf) algorithm that computes O(dg(p))O(d_g(p)) makespan solutions for k=2,3k = 2, 3 using mostly o(V2)o(|V|^2) running time. \paf provides worst case O(1)O(1)-approximation regarding solution time optimality. We note that the worst case running time for the problem is Ω(V2)\Omega(|V|^2).

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