Measuring Geometric Similarity Across Possible Plans for Automated Redistricting

Algorithmic and statistical approaches to congressional redistricting are becoming increasingly valuable tools in courts and redistricting commissions for quantifying gerrymandering in the United States. While there is existing literature covering how various Markov chain Monte Carlo distributions differ in terms of projected electoral outcomes and geometric quantifiers of compactness, there is still work to be done on measuring similarities between different congressional redistricting plans. This paper briefly introduces an intuitive and interpretive measure of similarity, and a corresponding assignment matrix, that corresponds to the percentage of a state's area or population that stays in the same congressional district between two plans. We then show how to calculate this measure in polynomial time and briefly demonstrate some potential use-cases.