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

31 January 2018
Jingjin Yu
ArXiv (abs)PDFHTML
Abstract

Let G=(V,E)G = (V, E)G=(V,E) be an m1×…×mkm_1 \times \ldots \times m_km1​×…×mk​ grid. Assuming that each v∈Vv \in Vv∈V is occupied by a robot and a robot may move to a neighboring vertex in a step via synchronized rotations along cycles of GGG, we first establish that the arbitrary reconfiguration of labeled robots on GGG can be performed in O(k∑imi)O(k\sum_i m_i)O(k∑i​mi​) makespan and requires O(∣V∣2)O(|V|^2)O(∣V∣2) running time in the worst case and o(∣V∣2)o(|V|^2)o(∣V∣2) when GGG is non-degenerate (i.e., nearly one dimensional). The resulting algorithm, \isag, provides average case O(1)O(1)O(1)-approximate (i.e., constant factor) time optimality guarantee. In the case when all dimensions are of similar size O(∣V∣1k)O(|V|^{\frac{1}{k}})O(∣V∣k1​), the running time of \isag approaches a linear O(∣V∣)O(|V|)O(∣V∣). Define dg(p)d_g(p)dg​(p) as the largest distance between individual initial and goal configurations over all robots for a given problem instance ppp, building on \isag, we develop the \pafalgo (\paf) algorithm that computes O(dg(p))O(d_g(p))O(dg​(p)) makespan solutions for k=2,3k = 2, 3k=2,3 using mostly o(∣V∣2)o(|V|^2)o(∣V∣2) running time. \paf provides worst case O(1)O(1)O(1)-approximation regarding solution time optimality. We note that the worst case running time for the problem is Ω(∣V∣2)\Omega(|V|^2)Ω(∣V∣2).

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