funsPDomNum2PE1D | R Documentation |
= 2
for Proportional Edge Proximity Catch Digraphs
(PE-PCDs) - middle interval caseThe function Pdom.num2PE1D
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
.
Pdom.num2AI(r, c, n)
Pdom.num2AII(r, c, n)
Pdom.num2AIII(r, c, n)
Pdom.num2AIV(r, c, n)
Pdom.num2A(r, c, n)
Pdom.num2Asym(r, c, n)
Pdom.num2BIII(r, c, n)
Pdom.num2B(r, c, n)
Pdom.num2Bsym(r, c, n)
Pdom.num2CIV(r, c, n)
Pdom.num2C(r, c, n)
Pdom.num2Csym(r, c, n)
Pdom.num2PE1D(r, c, n)
r |
A positive real number which serves as the expansion parameter in PE proximity region;
must be |
c |
A positive real number in |
n |
A positive integer representing the size of the uniform data set. |
P(
domination number\le 1)
for PE-PCD whose vertices are a uniform data set of size n
in a finite
interval (a,b)
Pdom.num2PE1D
The auxiliary functions are Pdom.num2AI, Pdom.num2AII, Pdom.num2AIII, Pdom.num2AIV, Pdom.num2A, Pdom.num2Asym, Pdom.num2BIII, Pdom.num2B, Pdom.num2B,
Pdom.num2Bsym, Pdom.num2CIV, Pdom.num2C
, and Pdom.num2Csym
, 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)
.
c \in ((3-\sqrt{5})/2, 1/2)
In Case A, we compute P(\gamma=2)
with
Pdom.num2AIV(r,c,n)
if 1 < r < (1-c)/c
;
Pdom.num2AIII(r,c,n)
if (1-c)/c< r < 1/(1-c)
;
Pdom.num2AII(r,c,n)
if 1/(1-c)< r < 1/c
;
and Pdom.num2AI(r,c,n)
otherwise.
Pdom.num2A(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 Pdom.num2Asym(r,c,n)
for c
in
(1/2,(\sqrt{5}-1)/2)
with the same auxiliary functions
Pdom.num2AIV(r,1-c,n)
if 1 < r < c/(1-c)
;
Pdom.num2AIII(r,1-c,n)
if (c/(1-c) < r < 1/c
;
Pdom.num2AII(r,1-c,n)
if 1/c < r < 1/(1-c)
;
and Pdom.num2AI(r,1-c,n)
otherwise.
c \in (1/4,(3-\sqrt{5})/2)
In Case B, we compute P(\gamma=2)
with
Pdom.num2AIV(r,c,n)
if 1 < r < 1/(1-c)
;
Pdom.num2BIII(r,c,n)
if 1/(1-c) < r < (1-c)/c
;
Pdom.num2AII(r,c,n)
if (1-c)/c < r < 1/c
;
and Pdom.num2AI(r,c,n)
otherwise.
Pdom.num2B(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 Pdom.num2Bsym(r,c,n)
for c
in
((\sqrt{5}-1)/2,3/4)
with the same auxiliary functions
Pdom.num2AIV(r,1-c,n)
if 1< r < 1/c
;
Pdom.num2BIII(r,1-c,n)
if 1/c < r < c/(1-c)
;
Pdom.num2AII(r,1-c,n)
if c/(1-c) < r < 1/(1-c)
;
and Pdom.num2AI(r,1-c,n)
otherwise.
c \in (0,1/4)
In Case C, we compute P(\gamma=2)
with
Pdom.num2AIV(r,c,n)
if 1< r < 1/(1-c)
;
Pdom.num2BIII(r,c,n)
if 1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c)
;
Pdom.num2CIV(r,c,n)
if (1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c)
;
Pdom.num2BIII(r,c,n)
if (1+\sqrt{1-4 c})/(2 c) < r <1/(1-c)
;
Pdom.num2AII(r,c,n)
if 1/(1-c) < r < 1/c
;
and Pdom.num2AI(r,c,n)
otherwise.
Pdom.num2C(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 Pdom.num2Csym(r,c,n)
for c \in (3/4,1)
with the same auxiliary functions
Pdom.num2AIV(r,1-c,n)
if 1< r < 1/c
;
Pdom.num2BIII(r,1-c,n)
if 1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c))
;
Pdom.num2CIV(r,1-c,n)
if (1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c))
;
Pdom.num2BIII(r,1-c,n)
if (1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c)
;
Pdom.num2AII(r,1-c,n)
if c/(1-c)< r < 1/(1-c)
;
and Pdom.num2AI(r,1-c,n)
otherwise.
Combining Cases A, B, and C, we get our main function Pdom.num2PE1D
which computes P(\gamma=2)
for any (r,c
) in its domain.
Elvan Ceyhan
Pdom.num2PEtri
and Pdom.num2PE1Dasy
#Examples for the main function Pdom.num2PE1D
r<-2
c<-.5
Pdom.num2PE1D(r,c,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)
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