PEdom.num.nondeg: The domination number of Proportional Edge Proximity Catch...

In pcds: Proximity Catch Digraphs and Their Applications

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple triangle case

Description

Returns the domination number, indices of a minimum dominating set of PE-PCD whose vertices are the data points in Xp in the multiple triangle case and domination numbers for the Delaunay triangles based on Yp points when PE-PCD is constructed with vertex regions based on non-degeneracy centers.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter r \ge 1 and vertex regions in each triangle are based on the center M which is one of the 3 centers that renders the asymptotic distribution of domination number to be non-degenerate for a given value of r in (1,1.5) and M is center of mass for r=1.5.

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). Loops are allowed for the domination number.

See (\insertCiteceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011,ceyhan:mcap2012;textualpcds) more on the domination number of PE-PCDs. Also, see (\insertCiteokabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

PEdom.num.nondeg(Xp, Yp, r)

Arguments

Xp	A set of 2D points which constitute the vertices of the PE-PCD.
Yp	A set of 2D points which constitute the vertices of the Delaunay triangles.
r	A positive real number which serves as the expansion parameter in PE proximity region; must be in (1,1.5] here.

Value

A list with three elements

dom.num

Domination number of the PE-PCD whose vertices are Xp points. PE proximity regions are constructed with respect to the Delaunay triangles based on the Yp points with expansion parameter r in (1,1.5].

#

mds	A minimum dominating set of the PE-PCD whose vertices are Xp points.
ind.mds	The data indices of the minimum dominating set of the PE-PCD whose vertices are Xp points.
tri.dom.nums	Domination numbers of the PE-PCD components for the Delaunay triangles.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

PEdom.num.tri, PEdom.num.tetra, dom.num.exact, and dom.num.greedy

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

r<-1.5  #try also r<-2

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

PEdom.num.nondeg(Xp,Yp,r)

pcds documentation built on May 29, 2024, 9:36 a.m.