One could use this process to draw legislative or congressional districts, through an iterative process.
Michigan has around 6600 election precincts, and each could be considered a potential candidate for one of Michigan's 14 congressional districts. Each iteration roughly 20% of the candidates would be eliminated. We can determined the number of iterations by solving for
x in the following equation.
6600 0.8
x = 14
x = (ln 14/6600) / ln .8
x = 27.59, which is rounded to the nearest integer, n, which is 28.
We can then solve for the actual decay rate,
d.
6600 d
n = 14
ln d = (ln 14/6600) / n
d = e
(ln 14/6600) / nd = .80264
We can easily calculate the target number,
t, for iteration
it
i= nint (6600 d
i)
t
1 = 5297
t
2 = 4252
...
t
10 = 732
...
t
81 = 81
...
t
28 = 14
We can reduce each of our 6600 proto-districts to a single point, their center of population, and then calculate a Voronoi diagram, and determine the center of population and total population of each.
We keep the t
1 (5297) most populous proto-districts, and eliminate the 6600 - 5297 (1303) least populous proto-districts, and repeat the process.
We can visualize the elimination of these smallest districts as somewhat analogous to the popping of soap bubbles, and the surviving neighboring bubbles expand to fill the space left by the eliminated soap bubble (proto-districts).
Let's say that the Salt Lake district in the above map is the least populous and were eliminated. The Denver, Las Vegas, Sacramento, Portland and Seattle districts would expand to fill the void. Note that we would have used the center of population for the districts, so the centers of the districts would have been somewhat different. The centers for Seattle, Portland, and Sacramento districts would have been pulled inward to begin with, and even further after Salt Lake were eliminated.
After 28 iterations we would have 14 districts of roughly equal population.
We can then adjust these districts by weighting the distance measure, so that less populous districts would expand towards more populous districts until the populations were equalized.
The survivng proto-districts in less densely populated areas will expand quickly, while in urban areas it will be slower.