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In representative democracy, a redistricting map is chosen to partition anelectorate into a collection of districts each of which elects arepresentative. A valid redistricting map must satisfy a collection ofconstraints such as being compact, contiguous, and of almost equal population.However, these imposed constraints are still loose enough to enable an enormousensemble of valid redistricting maps. This fact introduces a difficulty indrawing redistricting maps and it also enables a partisan legislature topossibly gerrymander by choosing a map which unfairly favors it. In this paper,we introduce an interpretable and tractable distance measure over redistrictingmaps which does not use election results and study its implications over theensemble of redistricting maps. Specifically, we define a central map which maybe considered as being "most typical" and give a rigorous justification for itby showing that it mirrors the Kemeny ranking in a scenario where we have acommittee voting over a collection of redistricting maps to be drawn. Weinclude run-time and sample complexity analysis for our algorithms, includingsome negative results which hold using any algorithm. We further study outlierdetection based on this distance measure. More precisely, we show gerrymanderedmaps that lie very far away from our central maps in comparison to a largeensemble of valid redistricting maps. Since our distance measure does not relyon election results, this gives a significant advantage in gerrymanderingdetection which is lacking in all previous methods.

Implications of Distance over Redistricting Maps: Central and Outlier Maps