Primary Research Sponsor Name: Bryan Quaife
Title: Professor
Research Supervisor Department: Scientific Computing
Contact Email: bquaife@fsu.edu

Project Title: Voting Districts and other Geometric Divisions
Project Website: 
Appropriate Major: Biology, Chemistry, Computer Science, Mathematics, 
  Political Science, Scientific Computing
DIS credit:  YES/NO
Number of assistants needed: 3-4
Project Duration: Fall and Spring
Is it a fall position: Yes
Is it a spring position: Yes
Is is a summer position: No

Description:

  Every 10 years, following the census, the United States Constitution 
specifies two procedures that adjust the House of Representatives in
the interests of fairness.  Reapportionment adjusts the total number of 
representatives assigned to each state, and is done on the basis of population, 
along with the requirement that every state must have at least one 
representative.  After a state has been assigned a given number of 
representatives, redistricting divides the state into congressional districts 
of roughly equal population.

  Reapportionment is a mathematical procedure.  However,
each state's portion of the population will be a decimal value with a fractional
part while the state can only get a whole number of representatives, so
some kind of rounding is required, and it turns out this is hard to do
in a fair way.  For example, in the 1880 reapportionment, Alabama was going 
to get 8 representatives.  It was then proposed to increase the total number 
of representatives, but it turned out that in that case, Alabama's share
would actually decrease to 7 representatives.  Every proposed method of 
reapportionment seems to display these kinds of strange behaviors.

  Redistricting is a geometric procedure, since it
requires starting with a map of the state, and a detailed population table,
and then drawing districts of the appropriate size.  Obviously, there is
great freedom in drawing these districts, and incumbents quickly discovered
that they could guarantee their own reelection if they could control how
their own district was determined.  Each state has passed laws that attempt
to control, reduce or even eliminate the ability of elected officials to
influence their districts, but controversy is almost always guaranteed.
Florida's congressional districts, corresponding to the 2010 census,
were the subject of a lawsuit in 2012, after which the state legislature
proposed several new district maps.  The state supreme court rejected
these maps in favor of one proposed by the plaintiffs.  This map, in
turn, is already the subject of further lawsuits.

  Redistricting is an interesting and practical example of a class of
problems involving the subdivision of a geometric region.  It is often
the case that many things must share a common space, such as trees in
a forest, or birds flying in a flock, or cells in a biological tissue.
These things want to be somewhat close, but not too close, so they
seem to cluster together while maintaining a certain distance.  These
examples from nature suggest that there may be ways to automatically
subdivide geometric regions into districts, or to cluster individual
objects without letting them get too close.  Our actual congressional
redistricting problem is a little more complicated because of the
fact that the population density varies throughout the state, but it
turns out there are natural ways to account for that as well.

  Ideas that will be useful for this problem include the history of the
proposals to reapportion the House of Representatives, the history of
redistricting controversies and the attempts to regularize the process,
numerical measurements of the quality of a clustering or redistricting,
the concepts of the Voronoi diagram, the Delaunay triangulation, and the
centroidal Voronoi tessellation or "CVT".

For informal discussions of the basic redistricting problem, you may 
refer to Nick Berry's article on Gerrymandering, at 
http://datagenetics.com/blog/march12015/index.html, or to Brian Hayes's
article on "Machine Politics" at 
http://bit-player.org/wp-content/extras/bph-publications/AmSci-1996-11-Hayes-redistrict/compsci96-11.pdf

For some insight into the mathematics behind the geometric problem,
you may want to look up the article:

   "Centroidal Voronoi Tessellations: Applications and Algorithms", 
   Qiang Du, Vance Faber, Max Gunzburger, 
   SIAM Review, 
   Volume 41, Number 4, December 1999, pages 637-676.

Task:

Some computer work will be needed, especially for the geometric studies.
Therefore, students will be expected to learn a suitable programming language,
in order to be able to read data files, calculate results, and produce
illustrations of the geometric subdivisions.

Students will familiarize themselves with the various reapportionment
strategies, create computer implementations of them, and compare them
on some actual reapportionment situations, such as the 1880 case which
disclosed the Alabama paradox, producing a short report.

Students will learn some of the concepts of computational geometry,
including clustering and subdivision.  They will experiment with the
determination of Voronoi diagrams and Delaunay triangulations.  They
will implement a computer procedure for producing a CVT of various
simple geometric shapes.

Students will choose a particular state, such as Florida or North Carolina,
where redistricting controversies have arisen.  They will gather information
about the geometric shape of the districts, and the population density.
They will estimate how closely the districts come to having equal populations.
Then they will investigate procedures for computing voting districts automatically,
using the CVT approach.