pcds (version 0.1.8)

Idom.numCSup.bnd.tri: Indicator for an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - one triangle case

Description

Returns \(I(\)domination number of CS-PCD is less than or equal to k\()\) where the vertices of the CS-PCD are the data points Xp, that is, returns 1 if the domination number of CS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of CS-PCD.

CS proximity region is constructed with respect to the triangle tri\(=T(A,B,C)\) with expansion parameter \(t>0\) and edge regions are based on the center \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of tri; default is \(M=(1,1,1)\) i.e., the center of mass of tri.

Edges of tri, \(AB\), \(BC\), \(AC\), are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph.

See also (ceyhan:mcap2012;textualpcds).

Caveat: It takes a long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.numCSup.bnd.tri(Xp, k, tri, t, M = c(1, 1, 1))

Value

A list with two elements

domUB

The upper bound k (to be checked) for the domination number of CS-PCD. It is prespecified as k in the function arguments.

Idom.num.up.bnd

The indicator for the upper bound for domination number of CS-PCD being the specified value k or not. It returns 1 if the upper bound is k, and 0 otherwise.

ind.domset

The vertices (i.e., data points) in the dominating set of size k if it exists, otherwise it is NULL.

Arguments

Xp

A set of 2D points which constitute the vertices of CS-PCD.

k

A positive integer to be tested for an upper bound for the domination number of CS-PCDs.

tri

A \(3 \times 2\) matrix with each row representing a vertex of the triangle.

t

A positive real number which serves as the expansion parameter in CS proximity region in the triangle tri.

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle tri; default is \(M=(1,1,1)\), i.e. the center of mass of tri.

Author

Elvan Ceyhan

References

See Also

Idom.numCSup.bnd.std.tri, Idom.num.up.bnd, Idom.numASup.bnd.tri, and dom.num.exact

Examples

# \donttest{
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-.5

Idom.numCSup.bnd.tri(Xp,1,Tr,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.tri(Xp,k,Tr,t,M)))
# }