The function PG2PE1D and its auxiliary functions.
Returns \(P(\gamma=2)\) for PE-PCD whose vertices are a uniform data set of size n in a finite interval
\((a,b)\) where \(\gamma\) stands for the domination number.
The PE proximity region \(N_{PE}(x,r,c)\) is defined with respect to \((a,b)\) with centrality parameter \(c \in (0,1)\) and expansion parameter \(r \ge 1\).
To compute the probability \(P(\gamma=2)\) for PE-PCD in the 1D case,
we partition the domain \((r,c)=(1,\infty) \times (0,1)\), and compute the probability for each partition
set. The sample size (i.e., number of vertices or data points) is a positive integer, n.
PG2AI(r, c, n)PG2AII(r, c, n)
PG2AIII(r, c, n)
PG2AIV(r, c, n)
PG2A(r, c, n)
PG2Asym(r, c, n)
PG2BIII(r, c, n)
PG2B(r, c, n)
PG2Bsym(r, c, n)
PG2CIV(r, c, n)
PG2C(r, c, n)
PG2Csym(r, c, n)
PG2PE1D(r, c, n)
A positive real number which serves as the expansion parameter in PE proximity region; must be \(\ge 1\).
A positive real number in \((0,1)\) parameterizing the center inside int\(=(a,b)\).
For the interval, int\(=(a,b)\), the parameterized center is \(M_c=a+c(b-a)\).
A positive integer representing the size of the uniform data set.
\(P(\)domination number\(=2)\) for PE-PCD whose vertices are a uniform data set of size n in a finite
interval \((a,b)\)
The auxiliary functions are PG2AI, PG2AII, PG2AIII, PG2AIV, PG2A, PG2Asym, PG2BIII, PG2B, PG2B,
PG2Bsym, PG2CIV, PG2C, and PG2Csym, each corresponding to a partition of the domain of
r and c. In particular, the domain partition is handled in 3 cases as
CASE A: \(c \in ((3-\sqrt{5})/2, 1/2)\)
CASE B: \(c \in (1/4,(3-\sqrt{5})/2)\) and
CASE C: \(c \in (0,1/4)\).
In Case A, we compute \(P(\gamma=2)\) with
PG2AIV(r,c,n) if \(1 < r < (1-c)/c\);
PG2AIII(r,c,n) if \((1-c)/c< r < 1/(1-c)\);
PG2AII(r,c,n) if \(1/(1-c)< r < 1/c\);
and PG2AI(r,c,n) otherwise.
PG2A(r,c,n) combines these functions in Case A: \(c \in ((3-\sqrt{5})/2,1/2)\).
Due to the symmetry in the PE proximity regions, we use PG2Asym(r,c,n) for \(c\) in
\((1/2,(\sqrt{5}-1)/2)\) with the same auxiliary functions
PG2AIV(r,1-c,n) if \(1 < r < c/(1-c)\);
PG2AIII(r,1-c,n) if \((c/(1-c) < r < 1/c\);
PG2AII(r,1-c,n) if \(1/c < r < 1/(1-c)\);
and PG2AI(r,1-c,n) otherwise.
In Case B, we compute \(P(\gamma=2)\) with
PG2AIV(r,c,n) if \(1 < r < 1/(1-c)\);
PG2BIII(r,c,n) if \(1/(1-c) < r < (1-c)/c\);
PG2AII(r,c,n) if \((1-c)/c < r < 1/c\);
and PG2AI(r,c,n) otherwise.
PG2B(r,c,n) combines these functions in Case B: \(c \in (1/4,(3-\sqrt{5})/2)\).
Due to the symmetry in the PE proximity regions, we use PG2Bsym(r,c,n) for c in
\(((\sqrt{5}-1)/2,3/4)\) with the same auxiliary functions
PG2AIV(r,1-c,n) if \( 1< r < 1/c\);
PG2BIII(r,1-c,n) if \(1/c < r < c/(1-c)\);
PG2AII(r,1-c,n) if \(c/(1-c) < r < 1/(1-c)\);
and PG2AI(r,1-c,n) otherwise.
In Case C, we compute \(P(\gamma=2)\) with
PG2AIV(r,c,n) if \(1< r < 1/(1-c)\);
PG2BIII(r,c,n) if \(1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c)\);
PG2CIV(r,c,n) if \((1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c)\);
PG2BIII(r,c,n) if \((1+\sqrt{1-4 c})/(2 c) < r <1/(1-c)\);
PG2AII(r,c,n) if \(1/(1-c) < r < 1/c\);
and PG2AI(r,c,n) otherwise.
PG2C(r,c,n) combines these functions in Case C: \(c \in (0,1/4)\).
Due to the symmetry in the PE proximity regions, we use PG2Csym(r,c,n) for \(c \in (3/4,1)\)
with the same auxiliary functions
PG2AIV(r,1-c,n) if \(1< r < 1/c\);
PG2BIII(r,1-c,n) if \(1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c))\);
PG2CIV(r,1-c,n) if \((1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c))\);
PG2BIII(r,1-c,n) if \((1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c)\);
PG2AII(r,1-c,n) if \(c/(1-c)< r < 1/(1-c)\);
and PG2AI(r,1-c,n) otherwise.
Combining Cases A, B, and C, we get our main function PG2PE1D which computes \(P(\gamma=2)\)
for any (r,c) in its domain.
PG2PEtri and PG2PE1D.asy
# NOT RUN {
#Examples for the main function PG2PE1D
r<-2
c<-.5
n<-10
PG2PE1D(r,c,n)
PG2PE1D(r=1.5,c=1/1.5,n=100)
# }
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