Until recently, gerrymandered districts tended to stick out, identifiable by their contorted tendrils. This is no longer the case. Without the telltale sign of an obviously misshapen district to go by, mathematicians have been developing increasingly powerful statistical methods for finding gerrymanders. These work by comparing a map to an ensemble of thousands or millions of possible maps. If the map results in noticeably more seats for Democrats or Republicans than would be expected from an average map, this is a sign that something fishy might have taken place.

But making such ensembles is trickier than it sounds, because it isn’t feasible to consider all possible maps — there are simply too many combinations for any supercomputer to count. A number of recent mathematical advances suggest ways to navigate this impossibly large space of possible simulations, giving mathematicians a reliable way to tell fair from unfair.

One advance came in 2019, when a group of researchers was working on a better way to draw a new district map for the Virginia House of Delegates. The previous year, a federal court had ruled that 11 districts in Virginia’s map were unconstitutional because they concentrated Black residents in a way that diluted their voting strength. Furthermore, Virginia has an unusually strict constraint in its redistricting process: Districts can only deviate in population by 1%. Given that there are 100 state House districts, “that’s a pretty tight bound,” said Daryl DeFord, a mathematician at Washington State University who analyzed the fairness of the Virginia map. It meant that the group couldn’t work at the level of precincts. “Some precincts were basically too big to make a valid plan,” DeFord said. Partitioning the map into smaller census block units didn’t work either. After around 10 million steps, standard flip-based MCMC algorithms “weren’t anywhere close to having representative samples from the whole space,” he said.

So DeFord and his colleagues came up with a way to move through the space more quickly. In order to speedily obtain samples from the entire space of possible maps, they needed to change the district assignment for many precincts at once in a way that preserved the contiguity of the districts. This made each step in the Markov chain more computationally expensive, but it also meant that each step brought them that much closer to the mixing time.

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