# [EM] Re: Compactness

Ernest Prabhakar drernie at mac.com
Mon Jan 19 09:16:11 PST 2004

```Hi Joe,

On Jan 18, 2004, at 8:42 PM, Joe Weinstein wrote:
> However, various writers on districting prefer instead cutely to use
> an extrinsic and sometimes more complex measure - boundary length.
> Hitherto, out of a certain feeling of charity, I have tended to regard
> this substitution as innocuous and acceptable.

I can appreciate that you may consider compactness a more aesthetic and
politically appropriate choice of district definition.   However, I'm
not sure why you consider it less complex and more easily computable
than boundary measures.

> (3)  For the sakes both of conceptual simplicity and of practical
> computation of a districting plan’s ‘goodness’ or ‘merit’, it’s
> preferable to define compactness or any other amenable concept
> intrinsically rather than extrinsically, whenever we can manage to do
> so.  In particular, it’s a lot simpler to compute just the average,
> over all SINGLE districts, of the district’s intrinsic sprawl, as
> versus having to compute the average, over all PAIRS of districts, of
> the weighted boundary between the pair.

Actually, in my experience, the result is actually the opposite.   Of
course, I define boundary minimization as minimizing the boundary of
each district, so I have no idea why one would need to calculate PAIRS.
Specifically, for n districts, I can calculate the boundary in order
(sqrt[n]), since I only  have to sum along the 'edge' districts.   This
is also amenable to a straightforward iterative process for improving
district boundaries,  since it is trivial to measure the impact on edge
length moving one atomic unit from district A to district B.

To do this using a compactness measure requires summing over all
districts seems to be a computation of order[n], which would be
costlier than the boundary measure.   Also, this doesn't lead itself
well to at least a naive minimization solution, since I believe adding
-any- units to a district would -always- decrease compactness, by most
measures.

Put another way, I've already come up with a fairly concise and
efficient way to implement a boundary minimization algorithm, which I
hope to post in the next day or two.  It seems to me than any
compactness based measures as you describe would actually be harder and
more time-consuming to calculate.  Of course,  perhaps you have a
clever algorithm for generating such districts that I'm not aware of,
which I'd be very interested in seeing.

Yours,
Ernie P.

> Joe Weinstein
>
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```