Geoscience Reference
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of the district area to the area of the smallest enclosing circle, while the latter
determines the ratio of the districts perimeter length to the circumference of a circle
with equal area
cmp
.D
k
/
D
A.D
k
/
r
enc
P.D
k
/
2
p
A.D
k
/
and
cmp
.D
k
/
D
;
where A.
/ and P.
/ denote the area and the length of the perimeter, respectively,
of a district and r
enc
the radius of the smallest enclosing circle (Young
1988
). For
the Reock (Schwartzberg) test, larger (smaller) ratios indicate greater compactness.
Other measures relate the activity of a district with the total activity of all basic
units within the smallest enclosing circle (Ricca and Simeone
2008
) or determine
the ratio of the squared diameter of a district and its area (Garfinkel and Nemhauser
1970
). A common global measure for the compactness of a districting plan is based
on the length of the boundary between districts, i.e., the total length of the perimeter
of the districts in the interior (Bozkaya et al.
2003
; Lei et al.
2012
)
p
X
cmp
.
D
/
D
P.D
k
/
P.J/:
kD1
Short inter-district boundaries typically result in compact districts. Numerous other
measures have been discussed in the literature. Unfortunately, none of them is
comprehensive; some fail to detect districts that are obviously noncompact, others
assign a low rating to visibly compact districts (Niemi et al.
1990
;Hornetal.
1993
;
Williams
1995
).
To use geometric measures for basic units representing points or lines, one can try
to give “shape” to the districts, for example through the smallest enclosing rectangle
or circle, or through the convex hull. Instead of the convex hull, one can also use -
shapes, which are polygons enclosing the point set that can provide a better fit to the
points than the convex hull (Duckham et al.
2008
). However, much more common
are the following, distance-based measures:
23.4.4.2
Distance-Based Measures
Distance-based measures are used predominantly in applications where people
have to travel within the districts, e.g., sales- or mailmen. This confers with the
motivation of compact districts in these applications: to reduce the day-to-day travel
times. Moreover, in these applications basic units typically represent points or lines,
making geometric measures unapplicable in the first place. The most common group
of local measures is based on the sum of distances between the center of a district
