Robotic fiber positioner (RFP) arrays are becoming heavily adopted in wide-field massively multiplexed spectroscopic survey instruments. RFP arrays decrease nightly operational overheads through rapid reconfiguration between fields and exposures. In comparison to similar instruments, SDSS-V has selected a very dense RFP packing scheme where any point in a field is typically accessible to three or more robots. This design provides flexibility in target assignment. However, the task of collisionless trajectory planning is especially challenging. We present two multiagent distributed control strategies that are highly efficient and computationally inexpensive for determining collision-free paths for RFPs in heavily overlapping workspaces. We demonstrate that a reconfiguration path between two arbitrary robot configurations can be efficiently found if a "folded" state, in which all robot arms are retracted and aligned in a lattice-like orientation, is inserted between the initial and final states. Although developed for SDSS-V, the approach we describe is generic and thus applicable to a wide range of RFP designs and layouts. Robotic fiber positioner technology continues to advance rapidly, and in the near future ultra-densely packed RFP designs may be feasible. Our algorithms are especially capable in routing paths in very crowded environments, where we see efficient results even in regimes significantly more crowded than the SDSS-V RFP design.