Revisiting Compactness for District Plans

TL;DR

The paper addresses the tension between shape-based and discrete compactness scores in redistricting by introducing a population-density conformal factor $\phi=\frac{P}{A}$ to define intermediate scores $\tilde{\mathcal{L}}$ and $\tilde{\mathcal{L}}_{lin}$ that interpolate between the total perimeter $L$ and the discrete cut-edge count $|\mathcal{C}|$. It also proposes a ReCom modification using edge-weighted spanning trees to bias the ensemble toward shorter, straighter boundaries via choices such as perimeter and slack, with county-based weighting explored as well. The key contributions include formal definitions of the conformal scores, conformal variants of Polsby-Popper and Schwartzberg, and multiple practical ReCom variants demonstrated on NC and Florida maps, showing improved shape-based compactness—particularly as the number of districts grows. Together, these results provide a principled, scalable framework for generating population-aware, geometry-respecting redistricting ensembles with potential legal and fairness implications.

Abstract

Modern sampling methods create ensembles of district maps that score well on discrete compactness scores, whereas the Polsby-Popper and other shape-based scores remain highly relevant for building fair maps and litigating unfair ones. The aim of this paper is twofold. First, we introduce population-weighted versions of shape-based scores and show a precise sense in which this interpolates between shape-based and discrete scores. Second, we introduce a modification of the ReCom sampling method that produces ensembles of maps with improved shape-based compactness scores.

Figures (9)

• Figure 1: The 2692 VTDs of North Carolina with county lines shown in red.
• Figure 2: The perimeter $l$ and Euclidean distance $d$ of the curve along which two precincts intersect. The slack$\mathfrak{s} = \frac{l}{d}\in[1,\infty)$ is a scale-invariant measurement of how wiggly the boundary is.
• Figure 3: The waist $\mathfrak{w}$ of three regular lattices.
• Figure 4: $\tilde{\text{PP}}$ verses $\text{PP}$ for the maps in standard ReCom ensembles.
• Figure 5: The kde plots of several compactness scores for ReCom NC congressional ensembles built with four choices of $\lambda$. Higher scores are better for $\text{PP}$ and $\tilde{\text{PP}}$, while lower scores are better for the other measurements.