Odd and Even

[jimrtex]

This map illustrates an apportionment of 7 x 1024 units (or atoms) among Alabama counties.  We could combine 1024 atoms to create each congressional district.  1024 = 2¹⁰, so we could have 10 rounds of pairwise combinations of increasingly larger groups.  In the final round we combine the 14 remaining combinations, each with a weight of 512 into the final 7 districts.

If we wanted to actually create the individual atoms we could subdivide each county.  Doing a subapportionment helps insure that any apportionment errors cancel out, since most early combinations would be within a county.

For the first round, the ideal population of each unit is approximately, 666.816   Using St.Lague apportionment we would expect the apportionment error to range from about half of this.  Assuming a uniform distribution of the error, there would be a mean (absolute) deviation of 167.  In actuality it is 190.

But instead of making the subapportionment, we will combine counties (or rather exchange population between counties).   We will choose pairs of (hopefully) adjacent counties with an odd number of units to do this. 

There are necessarily an even number of counties with an odd number of units, so we can always form pairs of counties.  We are not guaranteed that they will be adjacent, but in practice it is usually possible to do so.



Most of the pairings are forced.  For example, Sumter-Choctaw and Mobile-Baldwin are forced, this in turn forces Clarke-Monroe, then Escambia-Covington, then Lowndes-Butler.  In the southern group of 16 odd-unit counties, the only alternative would be to pair Crenshaw-Coffee, and Pike-Dale rather than Crenshaw-Pick and Coffee-Dale.  The latter, more natural pairing was adopted.

There is one isolated county, Chambers.  We pair it with Elmore across Tallapoosa, essentially forming two related pairs (Elmore-Tallapoosa and Tallapoosa-Chambers).  We could also pair Chambers with Clay or Coosa, but both of these would force a pairing of Chilton and Elmore.

[jimrtex]

We could simply merge pairs of odd-unit counties.   For example we could merge Lauderdale (139 units) and Lawrence (51 units), to create a new proto-district with 190 units.  We could then double the quota, such that the proto-district has a population of 95 units.  Counties with a an even-number of units, would simply have their number of units halved.  For example, Limestone would be reduced from 124 units (of 667) to 62 units (of 1334).

This would gradually reduce the number of proto-districts (25% decrease per round), and their number of units (33% per round).   But this is too quick a decrease in the number of proto-districts, so that there would less than 7 districts, and too slow a decrease in the population of districts, so that they would have a population greater than the final population.

So instead we instead make a minimal adjustment between pairs of odd-unit counties.  For example Lauderdale with 139 units and Lawrence with 51 units (of 667) could transfer one unit between them.  This would make both an even number (140 and 52; or 138 and 50 and the remnant could then be halved).  Once a county was reduced to one unit, it could start merging with adjacent counties/proto-districts.

But we can do slightly better by reducing the apportionment error as well.

The population of two counties with a odd-number of atoms is:

p₁ = (2m₁ + 1) q  + e₁
p₂ = (2m₂ + 1) q  + e₂

Where p_i is their population; m_i is an integer; q is the ideal population of an atom; and e_i is the apportionment error.

Because the rounding to an integer number of units is not independent (among the set of all counties), the error can be slightly greater than half of the apportionment unit.

q/2 ~> e_i ~> -q/2

In Alabama, Marshall was entitled to 139.497 atoms, but was rounded up to 140, and has an error of -335.219, which is slightly greater (in absolute terms) than half of the quota of 666.816 (333.408).   This does not matter at all, and we will see that the relative error quickly converges towards zero.

For example, for Lauderdale with a population of 92,709 and 139 atoms.

92709 = (68 * 2 + 1) * 666.816 + 22.597

We can choose the indices for the two counties such that the error for the 1^st county is greater (more positive) than that for the 2^nd county

q/2 ~> e₁ >= e₂ ~> -q/2

We can then arrange so that the error of both counties after the adjustment is the arithmetic mean of the two errors.

e_c = (e₁ + e₂)/2

The range of the errors will remain the same, but the distribution of errors will tend to converge towards zero.  If we assume a uniform distribution of errors after the initial apportionment, after the initial adjustment among pairs of odd-unit counties, the error will have a triangular distribution (and over many rounds converge to a normal distribution),

In particular, we can transfer:  

t = q - (e₁ - e_c)

or

t = q - (e₁ - e₂) / 2

From the 2^nd county to the 1^st county, such that the adjusted population of the two counties is:

r₁ = p₁ + t
r₂ = p₂ - t

r₁ = 2(m₁ + 1) q  + e_c
r₂ = 2m₂ q  + e_c

The adjusted population of both counties is now close to an even number of quotas, and the number of units can be cut in half, and remain an integer after the quota is doubled.  The transfer is small, less than the quota (by at least a small) bit and tends to converge the errors toward zero, and we have a consistent straightforward way to determine the direction of transfer.

Taking the specific pairing of Lauderdale and Lawrence:

Lauderdale   92,709 = 139 * 666.816 + 21.597
Lawrence   34,339 = 51 * 666.816 + 331.392

Since Lawrence has the greater error, we transfer 666.816 - (331.392 - 21.597)/2 persons from Lauderdale to Lawrence.   This calculates to 511.919 which we round to 512 persons.

Lauderdale (adjusted)    92,197 = 138 * 666.816 + 176.413
Lawrence (adjusted)   34,851 = 52 * 666.816 + 176.576

The adjusted population of the two counties is now near an even multiple of the quota.  We can double the quota and halve the apportionment:

Lauderdale (adjusted)    92,197 = 69 * 1333.632 + 176.413
Lawrence (adjusted)   34,851 = 26 * 1333.632 + 176.576

If we were actually creating 7 x 1024 districts of approximately equal population, in anticipation of pairwise merging into 7 x 512 districts, we would have one district crossing into Lauderdale County to capture 512 persons.   After merging with a district wholly in Lawrence, the (slight) majority of the merged district would be in Lawrence.

Alternatively, we could drive across the dam into Lauderdale County capture 512 nearby residents and have them vote in Lawrence County.

In practice, we simply make a book entry.  When we say "population of Lauderdale" we really mean 92,197 person (the actual population of 92,709 minus 512).  When we say "population of Lawrence" we really mean the actual population of 34,339 plus the 512 Lauderdale transfers.

[jimrtex]

We can then arrange so that the error of both counties after the adjustment is the arithmetic mean of the two errors.

e_c = (e₁ + e₂)/2

In particular, we can transfer:  

t = q - (e₁ - e_c)

or

t = q - (e₁ - e₂) / 2

From the 2^nd county to the 1^st county, such that the adjusted population of the two counties is:

r₁ = p₁ + t
r₂ = p₂ - t

r₁ = 2(m₁ + 1) q  + e_c
r₂ = 2m₂ q  + e_c

The adjusted population of both counties is now close to an even number of quotas, and the number of units can be cut in half, and remain an integer after the quota is doubled.  The transfer is small, less than the quota (by at least a small) bit and tends to converge the errors toward zero, and we have a consistent straightforward way to determine the direction of transfer.

We can extend this logic to the case where the odd-unit counties are not adjacent, and the transfer must be through an intermediate even-unit county.

That is the situation between Elmore and Chambers.   We must make two transfers, one between Elmore and Tallapoosa, and one between Tallapoosa and Chambers.  The adjusted population may increase by two units, stay the same, or decrease by two units, but it will remain an even number, while the odd-unit counties will be adjusted by one, to become even.

We want the error for each county to be the average of the that for the three counties: Elmore -48; Tallapoosa +273; Chambers +207, average 144.  Since Chambers has an error greater than the average, the transfer is from Tallapoosa towards Chambers:

t = q - (e_chambers - e_average).

And since Elmore has an error less than the average, its transfer is away from it towards Tallapoosa.

t = q - (e_average - e_elmore)

Population Before

Elmore: 119 q - 48
Tallapoosa: 62 q + 273
Chambers: 51 q + 207

transfer Tallapoosa to Chambers = 604
transfer Elmore to Tallapoosa = 474

Population after Adjustment

Elmore: 118 q + 145
Tallapoosa: 62 q + 143
Chambers: 52 q + 145

The errors are not equal since the transfers were rounded to a whole number of people, and the displayed errors have been truncated.

[jimrtex]

After the transfers (towards the black balls), all counties have a population approximately equivalent to an even number of 666.816 person units.  We can double the unit size, and halve the apportionment to each county, without any increase in error (it cuts the relative error in half).



Before the first round the error measurements were:

Standard Deviation: 213, Relative to Quota 32.0%.
Mean (Absolute) Deviation 190, Relative to Quota 28.5%.

Before the second round:

Standard Deviation 178, Relative To Quota 13.3%,
Mean (Absolute) Deviation 153, Relative to Quota 11.5%

As in the first round, we make balancing transfers between pairs of odd-unit counties:



The adjustment between Henry and Lee via Barbour and Russell, was chosen because it reduces extra transfers involving even unit counties.  The alternative of Henry-Dale-Coffee, would also require Pike-Montgomery-Autauga and Lee-Macon-Elmore.

As with three-county operations, we seek to balance the error between all four counties: Lee 216, Russell 193, Barbour 193, Dale 88 (average 172).  We have to transfer towards Lee and away from Dale.  After those transfers the error for Russell 237 and 109 for Barbour, which in turn dictates that the final transfer be from Barbour to Russell.

After the first round the adjusted population for Pike, included Pike 32,899 plus a book entry for Crenshaw of 506.  In the second round there is a transfer from Pike to Coffee of 1321.  We need to choose which population to transfer.

In general in a a transfer from County A (adjusted) to County B (adjusted), County A (adjusted) in addition to a large share of the actual County A population, may contain parts of County B, parts of common neighbors County C1, C2, ... and parts of non-common neighbors County D1, D2.

We given priority to returning any population from County B, then passing any County C1, etc. population on to County B.   The choice of which County C1, etc. is used would probably move the population that would minimize fragmenting these populations to a greater extent.  We can't pass on any from the non-common neighbors, so any final transfers are from County D.

[jimrtex]

This is the map after two rounds of transfers.   The standard deviation and mean absolute deviation are now 156 and 129, and 5.9% and 4.8% relative to a quota of 2667.3, which produces 256 x 7 protodistricts.

This produces the following pairing of odd-unit counties.



There is a major discretionary decision involving the 8-county ring of counties surrounding Cullman (Etowah is the 8th county).  The pairings can be rotated one position.   In addition, with the shown rotation, there could be pairings of Lauderdale-Lawrence and Limestone-Morgan.

I chose this particular configuration because it keeps the Decatur metropolitan area together, as well a bit better continuity of population between Jefferson and Blount, vs. Jefferson and Walker.

[jimrtex]

After 3 rounds, the standard deviation and mean absolute deviation are down to 142 and 120, respectively, and 2.7% and 2.2% respectively.  The number of persons displaced from their home counties (not shown) is 72.052 or about 1.5% of the state population.

Greene, the least populous county in Alabama, is the first county to reach 1 unit.  From now on, counties will begin to be merged into larger multi-county districts.

And here are the odd pairings.



Greene is to be merged with Sumter.  The odd-unit counties are relatively sparse (26 of 67 counties).   If county populations were random over a large range, it would be expected that about half would be odd, and have would be even. 

But we are compressing the range, and there appears to be a mild quantizing effect.   9 of the more rural counties have reached a population of 2 units (a population of grossly 11,000 persons).   Counties within a state have roughly the same area, and purely agricultural counties have comparable densities (their population is almost entirely controlled by the amount of arable land).   With about the same area and same densities, they will have similar populations and tend to be clustered.

[jimrtex]

This is after 4 rounds of pairwise-merging, or overall a 16:1 merging.



We have had our first merger of counties, but rather than Sumter+Greene, it is Choctaw+Sumter.  The transfer from Sumter to Choctaw, left the larger portion of Sumter with Choctaw, and left Greene with the less part of Sumter.

The standard deviation and mean absolute deviation are 139 and 117, or 1.30% and 1.10% relative to the quota of 10,669.  The total displaced population is 129,296 or 2.7% of the state population.

We are at the cusp of large scale merging of protodistricts.  11 districts have one unit, 16 have two units, and 6 have three units.   These 33 units (half of the total of 66) have a population just below that needed for one congressional district.  On the other hand, the 12 largest counties, those with more than 100,000 have a total of 269 units, or enough for slightly more than 4 districts.  It is this overall lumpiness that makes redistricting hard, like cutting a fruitcake without cutting any fruits, while the SCOTUS insist that each slice contain an identical number of fruit pieces.

Each eventual congressional district will contain 64 units, and we can begin to see an outline of possible districts.  Jefferson has almost enough population for 1 district; and Madison enough for half a district.   Add 5 counties (Jackson, Marshall, DeKalb, Etowah, and Cherokee) to its south, east, and southeast, and you have another district.  This then delimits a northwestern district that stops at Walker-Fayette-Lamar.

There is about enough population for a southeastern Montgomery-Houston district.  And Mobile-Baldwin need just a few more counties to reach enough for a district.

Here are the pairings for the 5th round.



There is a significant discretionary pairing in the southeast.  If Chilton-Coosa and Autauga-Elmore were replaced with Chilton-Autauga, Coosa-Clay, and Elmore-Montgomery, then there would be perhaps better pairing along the Georgia border: Randolph-Chambers. Lee-Russell, and Bullock-Pike.   I think I chose the set that I did based on the keeping the Montgomery suburbs together.

[jimrtex]

This is the map after 6 rounds as counties in the southern part of the state are beginning to merge.



The standard deviation is 152 and the mean absolute deviation is 127, an increase.  As counties are merged we can no longer average their errors, but rather their error is additive.

p₁ = q + e₁
p₂ = q + e₂

p_c = 2q + (e₁ + e₂)

As the quota doubles, the relative error continues to drop, the relative standard deviation to 0.71% and the relative mean absolute deviation to 0.60%.  The displaced population is now 239,431 or 5.0% of the state population.

Here are the odd pairings for the 6th round.

[jimrtex]

After 6 rounds or a 64:1 compression.



30 of the remaining 50 proto-districts are one unit districts.  The standard deviation is 161 and the mean absolute deviation is 132, an increase, with relative values of 0.38% and 0.31%, respectively.  The ratio of the standard deviation to the mean absolute deviation has increased slightly due to the merging of districts increasing some errors, while the errors for the larger counties remains low.  That is, the multi-county districts have somewhat greater errors.

388,655 or 8.1% of the population is displaced.

The final 7 congressional districts, each with a population of 16 units are fairly apparent now.

(1) Jefferson.
(2) Madison and 5 counties to the southeast.
(3) The northwest down to a Lamar-Fayette-Walker line.
(4) An east district between St.Clair-Calhoun-Cleburne and Chilton-Coosa-Tallapoosa-Chambers.
(5) Mobile-Baldwin-Escambia.

This leaves 32 units for the final 2 districts stretching from Tuscaloosa through Montgomery to Dothan, between districts (4) and (5).   Montgomery is somewhat in the middle, but it looks like it will be possible to have:

(6) Tuscaloosa to Montgomery
(7) Southeastern Alabama wrapping around Montgomery on both north and south.  Autauga and Elmore will be split apart.

With 36 of 50 proto-districts having a single unit, there was quite a bit discretion in pairings.  The pairings were chosen to help produce the above result and generally provide compactness.

[jimrtex]

After 7 rounds, we are down to 7 x 8 units (each congressional district will have 8 units).



28 of the 36 remaining districts are single unit, so we will see the areal extent expanding.  The standard deviation is 229 and the mean absolute value is 174, which are quite small relative to the unit size of 85,352 persons (0.27% and 0.20% respectively).

The number of displaced persons is 482,394 persons or 10.1% of the population.  These are shown along the county borders.  For example "Limestone" actually includes about 3 thousand persons from Lauderdale (2,600 actually).  So while we will be able to combine units with great accuracy (8 units is one congressional district) there is quite a bit of cross-border population that will have to be accounted for.

This shows the final districts.



The pairings were selected to bring this result into effect.

[jimrtex]

After 8 rounds.  Each unit is 1/4 of a congressional district.  If Alabama had 28 congressional districts it might look like this.



Standard deviation is 366, mean absolute deviation 277 (relative 0.21% and 0.16%).   Displaced population is 386,300 or 8.1%, which is a decline as neighboring districts merge.



These are the ninth round pairings.

[jimrtex]

After 9 rounds of merging we are down to 14 districts (2 in Jefferson).



The standard deviation is 601, mean absolute deviation 463.  Relative to the quota, they are 0.18% and 0.14%, respectively.  The displaced population is 206,978 or 4.3% of the total.  Almost half is between Baldwin and Mobile, which will be eliminated after the final mergers.



The final pairings.

[jimrtex]

This is after the final set of mergers.



The standard deviation is 989, with a mean absolute deviation of 824, with relative values of 0.14% and 0.12%, respectively.   Though small, it is somewhat surprising that it is this large.  The districts are largely comprised of whole counties, so that the error is the sum of the original errors at the county level which averaged around 200.  Normally, you would expect some more cancellation, but there was some rather surprising alignments.

For example, in the Tuscaloosa-Montgomery-Black Belt district, 14 of the 16 counties had a negative error, indicating a too large apportionment.   In the eastern district, 8 of 11 counties had a positive error.

At this point, it would be possible to equalize the population since there are displaced populations at most boundaries (a total of 84,336 or 1.8% of the population).

Instead, I eliminated the displaced populations that were less than 1% of the ideal district population.



This increases the standard deviation from 0.14% to 0.49%, while the displaced population decreases slightly from 84,338 to 71,038 (1.8% to 1.5%): Montgomery 29,213; Blount 23,970; and Lee 17,855.   The addition of about 24,000 from Blount or some other county to Jefferson is of course, barring a return of reason to the SCOTUS, is unavoidable.

We could swap Butler and either Lowndes or Wilcox for the Montgomery 29,000, though this would produce some relative erose borders.  Macon could be exchanged for the Lee 18,000 but that would cut off Elmore.  We are of course free to split either Elmore or Lee with no effect on equality.

Instead we do rotation and exchange:

West to East: Bibb
Southeast to West: Montgomery remnant - Butler
East to Southwest: Lee remnant.



This reduces the stranded population to that needed to get Jefferson up to the ideal population, but the standard deviation is up to 1.02%, as the three southern districts have increased their deviation, while eliminating the chops.

Moving Conecuh from the Southeast eliminates its surplus, and swapping Monroe for Clarke balances the other two districts.



This reduces the standard deviation to 0.38%, with 6 districts within 0.5% of the ideal, and the 7th within 1%.

[Sol]

This looks interesting, but I'm curious how this would work in states like AZ or MA, with fewer/less important counties.

[muon2]

This increases the standard deviation from 0.14% to 0.49%, while the displaced population decreases slightly from 84,338 to 71,038 (1.8% to 1.5%): Montgomery 29,213; Blount 23,970; and Lee 17,855.   The addition of about 24,000 from Blount or some other county to Jefferson is of course, barring a return of reason to the SCOTUS, is unavoidable.

Up to this point the process seems suitably algorithmic, though there seems to be no mechanism to accommodate the VRA. If I leave aside the VRA question the steps that follow this map seem somewhat more arbitrary. It looks like there are any number of swaps and rotations one could apply and no particular rule.

If I look to minimize population deviations with few county changes from the above plan, I would do the following shifts:

1. Swap Washington and Clarke which brings the Mobile district to Q (quota) -342.
2. Replace Chambers with Autauga in the Anniston district bringing it to Q -92.
3. Move Conecuh and Crenshaw into the Montgomery district bringing it to Q +357.
That leaves the Dothan district at Q +218.

The maximum deviation is now only 357 (0.05%) and the standard deviation drops to 230 (0.03%).

[jimrtex]

This increases the standard deviation from 0.14% to 0.49%, while the displaced population decreases slightly from 84,338 to 71,038 (1.8% to 1.5%): Montgomery 29,213; Blount 23,970; and Lee 17,855.   The addition of about 24,000 from Blount or some other county to Jefferson is of course, barring a return of reason to the SCOTUS, is unavoidable.

Up to this point the process seems suitably algorithmic, though there seems to be no mechanism to accommodate the VRA. If I leave aside the VRA question the steps that follow this map seem somewhat more arbitrary. It looks like there are any number of swaps and rotations one could apply and no particular rule.

If I look to minimize population deviations with few county changes from the above plan, I would do the following shifts:

1. Swap Washington and Clarke which brings the Mobile district to Q (quota) -342.
2. Replace Chambers with Autauga in the Anniston district bringing it to Q -92.
3. Move Conecuh and Crenshaw into the Montgomery district bringing it to Q +357.
That leaves the Dothan district at Q +218.

The maximum deviation is now only 357 (0.05%) and the standard deviation drops to 230 (0.03%).

Sigh!  And I thought I was doing so well.



Sheepishly offering my final map.  Concerned that the Mobile district was too low, I swapped Conecuh and Choctaw for Clarker, ending with a 0.30% standard deviation.



How do you weigh compactness against equality?

Regarding the VRA, it is arguable whether a plaintiff would satisfy the first prong of the Gingles tests, and with this district clearly fails the third prong.

For my version of the Tuscaloosa-Black Belt-Montgomery district:

2012 Obama 53.9% Romney 44.9%
2010 Sparks 53.0% Bentley 46.5%
2010 Barnes 47.6%, Shelby 52.4%
2008 Obama 52.6% McCain 46.8%
2008 Figures 49.8% Sessions 50.1%

[jimrtex]

This looks interesting, but I'm curious how this would work in states like AZ or MA, with fewer/less important counties.

In Massachusetts, towns could be used.   In Arizona, cities could be treated as separate from their counties.

[muon2]

This increases the standard deviation from 0.14% to 0.49%, while the displaced population decreases slightly from 84,338 to 71,038 (1.8% to 1.5%): Montgomery 29,213; Blount 23,970; and Lee 17,855.   The addition of about 24,000 from Blount or some other county to Jefferson is of course, barring a return of reason to the SCOTUS, is unavoidable.

Up to this point the process seems suitably algorithmic, though there seems to be no mechanism to accommodate the VRA. If I leave aside the VRA question the steps that follow this map seem somewhat more arbitrary. It looks like there are any number of swaps and rotations one could apply and no particular rule.

If I look to minimize population deviations with few county changes from the above plan, I would do the following shifts:

1. Swap Washington and Clarke which brings the Mobile district to Q (quota) -342.
2. Replace Chambers with Autauga in the Anniston district bringing it to Q -92.
3. Move Conecuh and Crenshaw into the Montgomery district bringing it to Q +357.
That leaves the Dothan district at Q +218.

The maximum deviation is now only 357 (0.05%) and the standard deviation drops to 230 (0.03%).

Sigh!  And I thought I was doing so well.



Sheepishly offering my final map.  Concerned that the Mobile district was too low, I swapped Conecuh and Choctaw for Clarker, ending with a 0.30% standard deviation.



How do you weigh compactness against equality?

This is a good question, and part of the answer can be the role of inequality in judging plans. Inequality can either be used as a primary or secondary factor.

If inequality is a primary factor then it should be considered equally with chops and erosity. That means a plan that has lower inequality can be kept for consideration, even if it is worse on either chops or erosity.

I think the threads have instead had the plans leaning towards using inequality as a secondary factor. That means that if there are two plans with equal chops and erosity then the plan with lower inequality eliminates the other plan. A plan with low inequality alone cannot replace a plan with a lower chop or erosity score.

Chops have by definition are a coarse measure. To compare erosity and inequality to chops on an equal footing I use a coarse scale for those as well. For example, our two plans don't differ that much on your perimeter score, and that is reflected in my erosity score of 48 for each plan. Since they each have one chop, they are tied on both primary measures. I would then turn to inequality as a potential tie breaker.

I use a fit to the range vs counties/districts that was compiled with the states drawn with no unneeded chops. The fit can be used directly as a logarithmic score giving your plan an inequality of 20 compared to my plan's score of 15. The fit can also be matched to the impact of chops by modeling the curve to an average state. In that case your plan scores 13 versus 4 for my plan. In either case the inequality in my plan would be measure less than that in your plan and with equal chops and erosity mine would be preferred.

[jimrtex]

Up to this point the process seems suitably algorithmic, [...snip VRA...] the steps that follow this map seem somewhat more arbitrary. It looks like there are any number of swaps and rotations one could apply and no particular rule.

It wasn't intended to be algorithmic, but rather interactive.

The idea had its genesis in Splitline, which has a tendency to produce needle-like districts.



This is because splitline uses the shortest median splitline (based on 180 orientations with a step of 180 degrees).  Splits can occur near corners, since this avoids crossing the entire area being divided.  In addition because medians tend to be through populated areas, can end up on the edges of areas from previous splits. 

If splitline used medians of area (adjusted to the nearest m:n-m population split, the splits would tend to be across the area between opposite sides, rather than between adjacent sides, which produces triangles, which are not compact.

Bur rather than doing this, it was suggest to do an additional level of splits, to produce twice as many districts as required, and then combine them based on longest adjacent sides.  So if you had two skinny areas that were then split crosswise, they would be recombined into a more compact form.


********************************************
*         A1      *           A2           *
*                 *                        *
********************************************
*          B1         *         B2         *
*                     *                    *
********************************************


So if A was divided into A1 and A2, and B into B1 and B2.  A1 and B1 might be joined into a final district, as would B1 and B2.

But if we could start from the whole state and slightly over-split and then recombine, why can't we start from the block level and combine areas?  We could apportion a population of 2²⁰ or 1,048,576 for each congressional district as our initial atom size which would have the effect of multiplying the actual population by 1.5 or so depending on the state.

Blocks or proto-districts with an odd number of units would be merged, and after division over 20 rounds would converge to a weight of one.  Conceivably you could let the odd blocks choose another odd block to merge with.  But you would have to deal with cases where there were not adjacent odd-blocks, or where a merger by neighbors left another block isolated.

You could deal with these by making corrections.  If an odd block was isolated, it could tunnel through to the nearest odd-block, making pairing adjustments on the way.

An alternative would be to use asymmetric merging, where the least populous districts were eliminated, with the voters choosing a new district.   This would not produce equal population districts, so another adjustment would be needed, but it would maximize voter choice.

An alternative would be to start with larger atoms.  They could be apportioned to counties on a statewide basis, and subsequently to townships and cities, and voting districts.

7x1024 units apportioned among Minnesota counties.



447 units for Anoka County apportioned among townships and cities, and then voting districts.



44 units for Becker County apportioned among townships and cities.  12 units were apportioned to Detroit Lakes, and subsequently among its 3 precincts.   Whole units were apportioned to larger townships and towns (those with the quota of 739 or more persons), mostly in the southern part of the county.   Fractional units of 1/16 were apportioned among the smaller townships, which were then merged into districts with one unit.



Not only would the apportionment be geographically stratified, but the merging would also be stratified.  For example, Anoka and Ramsey both have an odd number of units.  Townships/cities with an odd-number of units, and then precincts with an odd-number units within those two towns would be selected, and merged.  The area with the smaller population would be treated as if it had been annexed to the other town and county for subsequent rounds.   Elsewhere in all counties, there would be adjustments between odd unit townships/cities and mergers between odd-unit precincts.

The idea for using book entries in Alabama was derived from that in Minnesota where actual areas of adjacent counties would be treated as if annexed.  But rather than identifying actual territory, we simply keep track of how many persons would have been shifted.

[jimrtex]

One problem with pairing districts is that mandatory nesting of districts is not such a good idea.  Communities of interest don't come in quantum sized, let alone quantum sizes which are multiples of one another.

In Ohio, house districts may be quite reasonable, but when 3 are stuck together into a senate district they amy wiggle across several counties.  In Illinois, senate districts are sliced lengthwise like sausages.  If one is compact, the other isn't.

In California, where a reading of the constitution might lead to the belief that nesting of senate and assembly districts must be nested, the redistricting commission determined that it wasn't always such a good idea.

A particular problem with 2:1 nesting is that if a district is compact for one house, say a square, when combined with another square it will form a rectangle which is twice as long as it is wide, and if a square is cut in half, the two halves will also have a 2:1 ratio.   For districts which are not as compact, the combinations or divisions may be even worse.

So this is a test using a higher order nesting, specifically, five.  Five is somewhat of a compromise between four and seven.  If districts were square, then four districts could be combined to form a square.   If districts were hexagonal then seven districts could be combined into a roughly hexagonal shape (one of the districts and six neighbors).  I was also familiar with combining groups of districts based on their magnitude modulo 5, from my fractional representation maps.

The initial apportionment was into 5⁴ (625) times 7 congressional districts.  Units of approximately 1093 persons were apportioned among Alabama's counties.



For each county, we divide by 5 and take the remainder (ie number_of_units modulo 5).  So for example, Madison has 306 units, or 61x5 + 1 unit.  We can create 61 groups of 5 within Madison, and have one unit left over.  Similarly for Jackson, which has 49 units, we can create 9 groups of 5, with 4 units left over.  We can combine the extra units from the two counties to create another group of 5.

For Alabama, there are 120 extra units, so we must form 24 inter-county groups of 5.  Ideally, we would like to use simple pairings of adjacent counties like between Jackson and Madison, but that is not always possible.



In the above map there are 10 simple pairs.  There are also 2 groups of 5 units from 3 counties.  For example, Houston 3, Henry 1, and Dale 1 form a group.  We can think of the extra districts forming a district at the junction of the three counties, with about 3/5 in Houston, and 1/5 in each of the other two counties.

In other cases, groups of ten were formed.  For example, Lamar 3, Pickens 3, Tuscaloosa 3, and Fayette 1 have 10 extra.  We actually treat this as two groups of 5: Lamar 3, Fayette 1, and Pickens 1; and Tuscaloosa 3 and Pickens 2.  In essence the excess 3 units from Pickens will be split into 2 extra groups.

There are 6 groups of 10.   Technically, the group along the eastern border including Randolph, Coosa, Tallapoosa, Chambers, Elmore, Lee, and Russell is a group of 15, but it is treated as groups of 10 and 5, with the group of 10 linking through Tallapoosa, with some population from Lee shifting in, and some from Tallapoosa shifting out to Elmore.

The circles on the end of the links, indicate the direction of the shift.  For example the transfer is from Madison towards Jackson.

The actual number of persons transferred is adjusted so as to minimize the apportionment error among the counties, this is done by calculating the transfers so that after the transfers, the error for each county is close to the mean error among the counties (since transfers are rounded to whole persons, the errors are not precisely equal afterwards.

[jimrtex]

After the first round, there are a total of 875 units of 5463 persons left.  If Alabama had an 875 member legislature, this would be the apportionment among the counties, with some inter-county districts. not shown.  The apportionment error is rapidly decreasing, from a (relative) standard deviation of 29.0% to 3.3%.

25 of the 67 counties are apportioned less than 4 units, which means that counties will begin to be merged this round.



With 130 extra units there must be 26 groups of 5 created.  This is done with 9 groups of five, 3 groups of 10, and one group of 15, in the southeast corner of the state (Houston 4, Dale 4, Coffee 4, and Henry 3.

[jimrtex]

After 2 rounds we are down to 175 units of 27,313 persons each, with 25 units per eventual congressional district.   11 counties were merged into multi-county districts.  The apportionment error (standard deviation0 is down to 0.5%, from 29.0% and 3.3% before the first two rounds, at the expense of moving about 4% of the state population from their home counties.

Only the nine largest counties have five or more units, meaning that they will survive the next round with at least one unit.



There are an excess of 100 units, which must be grouped into 20 groups of 5.  This is less than the previous two rounds, due to the reduced number of districts (56 vs 67 counties), and the large number of single-unit districts.  Though there will be fewer groups, they will tend to be combining more districts, including potentially combining 5 single-unit districts.



In total, 16 groups of 5, and two groups of 10 are created.  Several are noteworthy.

The simple pairing of Elmore and Autauga has an important effect on the final result.  In addition, the creation of the grouping of Baldwin, Escambia, Conecuh, and Butler will have a significant effect.  This rather non-compact group was chosen to permit counties to be combined in the western part of the black belt.

The 10-unit grouping of Calhoun, Talladega, and St.Clair was treated as two five-units groups, which resulted in the significant splitting of St.Clair.  It would have been better to treat it as a single group of 10 units, to be merged into a 3-county 2-unit district.  

Jefferson ended up in a group with 5 counties to the west, since none of the adjacent counties had a single unit to combine with the 4 extra units from Jefferson.  The effect of this is that the final Jefferson-based district will extend into Walker County.

[jimrtex]

We are down to 35 units of 136,564, or 5 per congressional district.  If Alabama had 35 senate districts, they might look like this.  The population that is shifted in from adjacent districts is indicated in small letters and numbers, representing 1000s of persons.   For example the Jackson-DeKalb district includes roughly 11,000 persons from Marshall, and 1,000 from Madison.  Where possible, the shifted populations have been consolidated, so that between adjacent districts only one county has a shift.  

About 6.1% of the state population is shifted, with about half of that in 4 major shifts (Madison 60K, Baldwin 55K, St.Clair 28K, and Walker 22K).  Where possible, the final set of groupings will be done so as to eliminate these shifts.

The relative standard deviation is 0.2%.



We group the districts into the provisional final districts, with a particular effort to eliminate larger divisions of counties.  The northern part of the state aligns along lines found previously, while the southern part is simply to fit together anything feasible.

[jimrtex]

After the final merging of districts, we consolidate split populations where possible.  In doing so, a slim majority of Pike ends up in the Black Belt district.  The standard deviation is 0.09%.  Around 58,000 persons, or 1.22% of the state population are split across county lines, with 3/7 of this number needed to augment Jefferson to comprise a single district.



We disregard the smallest chops, and assume that we can have an equal split between the Northwest and Jefferson districts.  The standard deviation increases to 0.17%.



We finalize the map by eliminating the three small shifts in the south.  This is done with a counter-clockwise rotation in two steps.

(1) All of Monroe is returned to the Black Belt district, the southwestern district takes more of Covington, and the southeastern district takes the majority of Pike.  Since all shifts were identical to the Monroe population, the standard deviation remained 0.17%.

(2) Butler is transferred to the Black Belt, all of Covington to the southwestern district, and the remnant of Pike plus Crenshaw were transferred to the southeastern district.

In addition, an actual split of Walker on precinct lines was done.



These changes increased the standard deviation to 0.25%.   The perimeter is 906 miles, which I believe is the best done yet.

The BVAP of the Black Belt district is 45% with the following election results:

2008P Obama 53.7 McCain 45.8%
2008S Figures 51.0% Sessions 48.9%
2010G Sparks 54.4% Bentley 45.1%
2010S Barnes 49.0% Shelby 50.9%
2012P Obama 55.0% Romney 44.3%