Geoscience Reference
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where f
ij
is the flow from basic unit i to j and f
c
k
;j
D
0,
8
j
2
N.c
k
/ (Shirabe
2009
).
A simpler approach is to require that each district is a subtree of a shortest
path tree T.c
k
/ rooted at the district center c
k
, where the edge lengths typically
correspond to road distances or are all assumed to be 1. Then, for each basic unit
j of district k, at least one of the adjacent basic units i
2
N.j/ that immediately
precedes j on some shortest path to the center c
k
also has to be included in the
district:
x
kj
X
i2S
j
x
ki
8
j
2
J
nf
c
k
g
;
where S
j
Df
i
2
N.j/
j
i immediately precedes j on some shortest path from
j to c
k
g
(Zoltners and Sinha
1983
; Mehrotra et al.
1998
). Although this excludes
some contiguous districts, these are unlikely to be compact, as they typically have
large protrusions or indentations, or contain enclaves.
It is straight forward to extend all of the above constraints to the case where
the choice of district centers is part of the optimization. For geometric contiguity
measures obviously only informal mathematical formulations can be derived.
Remark 23.1
Only few authors try to derive approximate neighborhood graphs
for point-like basic units. The majority simply does not consider contiguity at all
and tries to obtain districts with little overlap through an appropriate compactness
measure, see also Example
23.3
.
23.4.4
Compactness
A district is said to be geographically compact if it is somewhat round-shaped and
undistorted. The motivation for compact districts is almost identical to ensuring
contiguity: to prevent gerrymandering or to reduce the day-to-day travel distances
within the districts. Although being a very intuitive concept, a rigorous definition
of compactness does not exist and, moreover, strongly depends on the geometric
representation of basic units. In the context of political districting, typically mea-
sures based on the shape of districts are employed whereas in sales and distribution
districting, distance-based measures are predominant. In the following, the most
common ones for both approaches are presented.
23.4.4.1
Geometric Measures
If basic units are given as polygons, geometric approaches based on the area or
perimeter of a district can be used to quantify compactness. Two common local
measures are the Reock and Schwartzberg tests. The former calculates the ratio
