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

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