A Novel Space Partitioning Algorithm to Improve Current Practices in Facility Placement Page: 9
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IEEE TRANSACTIONS ON SYSTEM, MAN, AND CYBERNETICS PART A, VOL. X, NO. X, MARCH 2011
a1 (p - bmax)+bmax
ala2 (P - bmax) + bmax
(p - bmax) ai + bmax
Q1 (P- bmax) -/3ip-bmazbi
p H /l
Fig. 6. General case: k PODs while k is not required to be a power of 2
This can be simplified by applying Lemma 3:
popmin - - P
b 2 h2 )
The right child of a particular node is at most assigned 2 of
the population, as the algorithm assigns the larger proportion
to the left child. Hence /3j < with j e [1..h2]. Therefore
POPrin > -P
By applying the geometric series this can be transformed into
the following inequality:
bmax 1 ( - h2
The maximum difference in populations sizes is calculated by
subtracting the minimum population size from the maximum
population size of a catchment area:
S- bmax (1 (2)h2
hi, from left to right. Since the recursion tree is not necessarily
complete, the height of the right side of the tree can either be
h2 1 hi [= log(k) or h2 L[log(k)]. If k is a power of 2, it
follows that hi = h2 = Llog(k)] = [log(k) = log(k). If k is
not a power of 2, then let k' denote the largest number smaller
than k that is a power of 2. This yields a height h2 = log(k').
Note that if k approaches infinity, k' also approaches infinity.
Case 1: h2 = h1 (complete tree)
h2 - h = log(k) k = 2h2 = 2h1
Amax < - bmax + bmax + bmax (1 - )
Sbmax + bmax + bmax (1 lgk)
Case 2: h2 - h1 (tree not complete)
h2 = hi -1 = log(k') = [log(k)] # k' = 2h2 2h=2-1
k' < k + log(k') < log(k)
bmax + bmax + bmax 1
Let hl denote the height of the left side of the tree, i.e. the
length of the path from the root node to the leftmost leaf. Then
h1 = [log(k)], as Algorithm 1 fills the last tree level, i.e. level
Amax < - bmax + bmax + bmax 1
Sbmx + bmax + b max 1 ( ()
/2/31P -/32/3lbr x-
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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/m1/9/: accessed May 29, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Engineering.