# [EM] Advantages of Dirichlet Region Districts

Forest Simmons fsimmons at pcc.edu
Thu Jan 15 12:40:07 PST 2004

```On Wed, 14 Jan 2004, Ernest Prabhakar wrote:

> Hi Forest,
>
> On Jan 14, 2004, at 2:26 PM, Forest Simmons wrote:
>
> > If voting districts were required to be Dirichlet regions (with
> > respect to
> > their "centers"), then ...
>
> Apologies if I missed it or, I'm being dense, but how do we create
> well-defined Dirichlet districts when there are multiple polling places
> per district?   This is even more complicated in the U.S., since we
> often have different polling places for different years (more in
> presidential, less when its just state).
>

I knew I shouldn't have mentioned that possibility! I knew it might give
the impression of inevitability rather than mere possibility :-)

The defining point or "center" of a Dirichlet style district doesn't have
to be a polling place, but it could be made into one, if it (the "center"
point) happened to be located in a room in a public building.

Here in Oregon every residence is a polling place; we vote by mail.

That said, due to other considerations I'm retracting my suggestion that
redistricting proposals be required to be Dirichlet style proposals; as
Joe Weinstein pointed out long ago, any partition of the voters into the
specified number of sets (proposed districts) should be allowed as a
proposal.

Proposals with districts that differ by too many voters would be
eliminated immediately. The remaining proposals would be evaluated for
over-all compactness by an objective measure.

[If that measure is the average within district distance between voters,
then those proposals based on Dirichlet regions would tend to have an

If there are n districts and N voters per district, then the number of
within district distances to average together would be

N*n*(n-1)/2 ,

(or twice this number if we take into account the existence of one way
ramps, lanes, streets, etc.) which is well within the reach of modern
computational power, especially since this kind of calculation is
eminently amenable to parallel computation.

In any case, a Montecarlo approximation (based on sufficiently many
randomly chosen within district pairs) would be adequate to single out the
finalists among the proposals, if not decide the whole contest.

As I mentioned previously, the MapQuest folks seem to have already solved
the travel distance computation problem, probably by considering the
network of roads as a directed network, and applying some version of
Djikstra's algorithm (familiar to our readers from the recent discussion
of Beatpath algorithms).

This travel distance is more appropriate than the "as the crow flies"
distance in this context, because it is more apt to preserve natural
boundaries of communities.

As I mentioned elsewhere most of the credit for this idea goes to Joe
Weinstein.  I don't remember exactly what measure he proposed for over-all
compactness, though it may have been the total boundary length.

The advantages of my proposed measure are ...

(1) It doesn't require drawing boundaries. A proposal is any well defined
partition of the voters into the specified number of sets. This can be
done in many ways without drawing boundaries, including submitting lists
of voters (the most extravagant extreme) and specifying Dirichlet centers
(near the most economical extreme).

(2) Natural boundaries such as rivers are more likely to be respected by
my within district average length measure than by the minimum boundary
length measure.

(3) My measure is easy to calculate no matter how the partition is
specified.  The boundary length measure might be difficult if not
impossible to apply to a proposal that was specified by a mere listing of
the voters comprising the proposed districts, for example.

Unless anybody has a better idea, I suggest that we start figuring out how
to get Joe's method adopted into law in some forward thinking state or
country (if there is such a place).

Or how about Afghanistan or Iraq?  Aren't they supposed to be the
showcases of U.S. imposed western democracy?

Let's put in our bid before Diebold is awarded the contract.

My Best,

Forest

```