Random sampling of graph partitions under
constraints has become a popular tool for evaluating legislative
redistricting plans. Analysts detect partisan gerrymandering by
comparing a proposed redistricting plan with an ensemble of sampled
alternative plans. For successful application, sampling methods
must scale to large maps with many districts, incorporate realistic
legal constraints, and accurately and efficiently sample from a
selected target distribution. Unfortunately, most existing methods
struggle in at least one of these areas. We present a new
Sequential Monte Carlo (SMC) algorithm that draws representative
redistricting plans from a realistic target distribution of choice.
Because it samples directly, the SMC algorithm can efficiently
explore the relevant space of redistricting plans better than the
existing Markov chain Monte Carlo (MCMC) algorithms that yield
dependent samples. Our algorithm can simultaneously incorporate
several constraints commonly imposed in real-world redistricting
problems, including equal population, compactness, and preservation
of administrative boundaries. We validate the accuracy of the
proposed algorithm by using a small map where all redistricting
plans can be enumerated. We then apply the SMC algorithm to
evaluate the partisan implications of several maps submitted by
relevant parties in a recent high-profile redistricting case in the
state of Pennsylvania. We find that the proposed algorithm is
roughly 40 times more efficient in sampling from the target
distribution than a state-of-the-art MCMC algorithm.
Open-source
software is available for implementing the proposed
methodology. (Last Revised, December, 2020)