A Novel Space Partitioning Algorithm to Improve Current Practices in Facility Placement Page: 7
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IEEE TRANSACTIONS ON SYSTEM, MAN, AND CYBERNETICS PART A, VOL. X, NO. X, MARCH 2011
P - bax+
Fig. 4. Maximum and minimum population sizes at each recursion level
cases yield [pop(CA2) - bmax,pop(CA2) - 1] and
[pop(CA2),pop(CA2) - 1 + bmax]. Hence, the maxi-
mum possible difference in population size between the
catchment areas is bmax.
2) pop(CA1) > pop(CA2): This case is analogous to
pop(CA1) < pop(CA2).
3) pop(CA1) = pop(CA2): Either of the catchment areas
is assigned the last census block blast. Without loss of
generality, assume that blast is assigned to CA1. With
pop(blast) E [0, bmax] the population of CA1 is then
in the range [pop(CA2), pop(CA2) + bmax]. Hence, the
maximum possible difference between the catchment
areas is bmax.
For all of the cases, the maximum possible population dif-
ference between the two catchment areas is bmax, and conse-
quently, Amax e [0..bmax]. Having established the maximum
difference in population size between two catchment areas, we
will need to bound the maximum population difference across
the entire partition of the geographic space for k > 2 PODs.
Theorem 1. For the placement of k = 2h PODs, a total
population size p and largest population size of a census block
bmax, Amax is bounded and Amax e [0..2bmax].
As stated by Lemma 2, the maximum difference of popula-
tion at each recursion level is bmax. During each recursive step,
Lemma 2 can be applied repeatedly. The resulting possible
maximum and minimum populations for different recursion
steps are shown in Figure 4. The left and right child of each
node differ by exactly bmax individuals. Note that the sum of
the population of all nodes at each level is p. The leftmost leaf
node of the tree contains the region with the highest possible
population size, whereas the rightmost leaf node of the tree
contains the region with the lowest possible population size.
1 1 1
5 5 5
(a) Portion of population
1 1 1
3 2 2
(b) Proportions assigned at each level
Fig. 5. Example of k = 5 PODs
The maximum possible population size pop,,,ax and the
minimum possible population size popin for a tree of height
h are calculated as follows:
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Jimenez, Tamara; Mikler, Armin R. & Tiwari, Chetan. A Novel Space Partitioning Algorithm to Improve Current Practices in Facility Placement, article, March 2011; [New York, New York]. (digital.library.unt.edu/ark:/67531/metadc132975/m1/7/: accessed February 19, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Engineering.