In representative democracy, a redistricting map is chosen to partition an electorate into a collection of districts each of which elects a representative. A valid redistricting map must satisfy a collection of constraints such as being compact, contiguous, and of almost equal population. However, these imposed constraints are still loose enough to enable an enormous ensemble of valid redistricting maps. This fact introduces a difficulty in drawing redistricting maps and it also enables a partisan legislature to possibly gerrymander by choosing a map which unfairly favors it. In this paper, we introduce an interpretable and tractable distance measure over redistricting maps which does not use election results and study its implications over the ensemble of redistricting maps. Specifically, we define a central map which may be considered as being "most typical" and give a rigorous justification for it by showing that it mirrors the Kemeny ranking in a scenario where we have a committee voting over a collection of redistricting maps to be drawn. We include run-time and sample complexity analysis for our algorithms, including some negative results which hold using any algorithm. We further study outlier detection based on this distance measure. More precisely, we show gerrymandered maps that lie very far away from our central maps in comparison to a large ensemble of valid redistricting maps. Since our distance measure does not rely on election results, this gives a significant advantage in gerrymandering detection which is lacking in all previous methods.

翻译：在代议制民主中，选择一张选区重划地图来将全体选民划分为若干选区，每个选区选举出一名代表。一张有效的选区重划地图必须满足一系列约束条件，例如紧凑性、连续性以及人口几乎相等。然而，这些强制约束仍然足够宽松，以至于允许存在海量有效的选区重划地图。这一事实给绘制选区重划地图带来了困难，同时也使存在党派偏见的立法机构可能通过选择一张不公平地有利于自己的地图来进行不公正的选区划分。在本文中，我们提出了一种可解释且易处理的选区重划地图间距离度量，该度量不使用选举结果，并研究了其在整个选区重划地图集合中的含义。具体而言，我们定义了一张可被视为“最典型”的中心地图，并通过证明在由委员会对一组待绘制的选区重划地图进行投票的场景中，该中心地图反映了Kemeny排序，为其提供了严格的合理性证明。我们分析了算法的运行时间和样本复杂度，包括任何算法都无法回避的一些负面结论。我们进一步研究了基于此距离度量的离群点检测。更准确地说，我们展示了与大量有效选区重划地图集合相比，远离中心地图的不公正划分地图。由于我们的距离度量不依赖于选举结果，这使得在不公正划分检测方面具有显著优势，这是以往所有方法所缺乏的。